analysis.normed_space.inner_product

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@[class]

structure has_inner (𝕜 : Type u_4) (E : Type u_5) :

Type (max u_4 u_5)

inner : E → E → 𝕜

Syntactic typeclass for types endowed with an inner product

Instances

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@[class]

structure inner_product_space (𝕜 : Type u_4) (E : Type u_5) [is_R_or_C 𝕜] :

Type (max u_4 u_5)

to_normed_group : normed_group E
to_normed_space : normed_space 𝕜 E
to_has_inner : has_inner 𝕜 E
norm_sq_eq_inner : ∀ (x : E), ∥x∥ ^ 2 = ⇑is_R_or_C.re (inner x x)
conj_sym : ∀ (x y : E), ⇑is_R_or_C.conj (inner y x) = inner x y
add_left : ∀ (x y z : E), inner (x + y) z = inner x z + inner y z
smul_left : ∀ (x y : E) (r : 𝕜), inner (r • x) y = (⇑is_R_or_C.conj r) * inner x y

An inner product space is a vector space with an additional operation called inner product. The norm could be derived from the inner product, instead we require the existence of a norm and the fact that ∥x∥^2 = re ⟪x, x⟫ to be able to put instances on 𝕂 or product spaces.

To construct a norm from an inner product, see inner_product_space.of_core.

Instances

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@[nolint, class]

structure inner_product_space.core (𝕜 : Type u_4) (F : Type u_5) [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] :

Type (max u_4 u_5)

inner : F → F → 𝕜
conj_sym : ∀ (x y : F), ⇑is_R_or_C.conj (c.inner y x) = c.inner x y
nonneg_re : ∀ (x : F), 0 ≤ ⇑is_R_or_C.re (c.inner x x)
definite : ∀ (x : F), c.inner x x = 0 → x = 0
add_left : ∀ (x y z : F), c.inner (x + y) z = c.inner x z + c.inner y z
smul_left : ∀ (x y : F) (r : 𝕜), c.inner (r • x) y = (⇑is_R_or_C.conj r) * c.inner x y

A structure requiring that a scalar product is positive definite and symmetric, from which one can construct an inner_product_space instance in inner_product_space.of_core.

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def inner_product_space.of_core.to_has_inner {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] :

has_inner 𝕜 F

Inner product defined by the inner_product_space.core structure.

Equations

inner_product_space.of_core.to_has_inner = {inner := c.inner}

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def inner_product_space.of_core.norm_sq {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] (x : F) :

ℝ

The norm squared function for inner_product_space.core structure.

Equations

inner_product_space.of_core.norm_sq x = ⇑is_R_or_C.re (inner x x)

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theorem inner_product_space.of_core.inner_conj_sym {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] (x y : F) :

⇑is_R_or_C.conj (inner y x) = inner x y

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theorem inner_product_space.of_core.inner_self_nonneg {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

0 ≤ ⇑is_R_or_C.re (inner x x)

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theorem inner_product_space.of_core.inner_self_nonneg_im {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

⇑is_R_or_C.im (inner x x) = 0

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theorem inner_product_space.of_core.inner_self_im_zero {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

⇑is_R_or_C.im (inner x x) = 0

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theorem inner_product_space.of_core.inner_add_left {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y z : F} :

inner (x + y) z = inner x z + inner y z

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theorem inner_product_space.of_core.inner_add_right {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y z : F} :

inner x (y + z) = inner x y + inner x z

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theorem inner_product_space.of_core.inner_norm_sq_eq_inner_self {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] (x : F) :

↑(inner_product_space.of_core.norm_sq x) = inner x x

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theorem inner_product_space.of_core.inner_re_symm {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

⇑is_R_or_C.re (inner x y) = ⇑is_R_or_C.re (inner y x)

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theorem inner_product_space.of_core.inner_im_symm {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

⇑is_R_or_C.im (inner x y) = -⇑is_R_or_C.im (inner y x)

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theorem inner_product_space.of_core.inner_smul_left {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} {r : 𝕜} :

inner (r • x) y = (⇑is_R_or_C.conj r) * inner x y

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theorem inner_product_space.of_core.inner_smul_right {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} {r : 𝕜} :

inner x (r • y) = r * inner x y

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theorem inner_product_space.of_core.inner_zero_left {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

inner 0 x = 0

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theorem inner_product_space.of_core.inner_zero_right {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

inner x 0 = 0

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theorem inner_product_space.of_core.inner_self_eq_zero {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

inner x x = 0 ↔ x = 0

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theorem inner_product_space.of_core.inner_self_re_to_K {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

↑(⇑is_R_or_C.re (inner x x)) = inner x x

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theorem inner_product_space.of_core.inner_abs_conj_sym {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

is_R_or_C.abs (inner x y) = is_R_or_C.abs (inner y x)

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theorem inner_product_space.of_core.inner_neg_left {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

inner (-x) y = -inner x y

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theorem inner_product_space.of_core.inner_neg_right {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

inner x (-y) = -inner x y

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theorem inner_product_space.of_core.inner_sub_left {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y z : F} :

inner (x - y) z = inner x z - inner y z

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theorem inner_product_space.of_core.inner_sub_right {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y z : F} :

inner x (y - z) = inner x y - inner x z

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theorem inner_product_space.of_core.inner_mul_conj_re_abs {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

⇑is_R_or_C.re ((inner x y) * inner y x) = is_R_or_C.abs ((inner x y) * inner y x)

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theorem inner_product_space.of_core.inner_add_add_self {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

inner (x + y) (x + y) = inner x x + inner x y + inner y x + inner y y

Expand inner (x + y) (x + y)

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theorem inner_product_space.of_core.inner_sub_sub_self {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x y : F} :

inner (x - y) (x - y) = inner x x - inner x y - inner y x + inner y y

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theorem inner_product_space.of_core.inner_mul_inner_self_le {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] (x y : F) :

(is_R_or_C.abs (inner x y)) * is_R_or_C.abs (inner y x) ≤ (⇑is_R_or_C.re (inner x x)) * ⇑is_R_or_C.re (inner y y)

Cauchy–Schwarz inequality. This proof follows "Proof 2" on Wikipedia. We need this for the core structure to prove the triangle inequality below when showing the core is a normed group.

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def inner_product_space.of_core.to_has_norm {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] :

has_norm F

Norm constructed from a inner_product_space.core structure, defined to be the square root of the scalar product.

Equations

inner_product_space.of_core.to_has_norm = {norm := λ (x : F), real.sqrt (⇑is_R_or_C.re (inner x x))}

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theorem inner_product_space.of_core.norm_eq_sqrt_inner {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] (x : F) :

∥x∥ = real.sqrt (⇑is_R_or_C.re (inner x x))

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theorem inner_product_space.of_core.inner_self_eq_norm_sq {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] (x : F) :

⇑is_R_or_C.re (inner x x) = ∥x∥ * ∥x∥

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theorem inner_product_space.of_core.sqrt_norm_sq_eq_norm {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] {x : F} :

real.sqrt (inner_product_space.of_core.norm_sq x) = ∥x∥

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theorem inner_product_space.of_core.abs_inner_le_norm {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] (x y : F) :

is_R_or_C.abs (inner x y) ≤ ∥x∥ * ∥y∥

Cauchy–Schwarz inequality with norm

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def inner_product_space.of_core.to_normed_group {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] :

normed_group F

Normed group structure constructed from an inner_product_space.core structure

Equations

inner_product_space.of_core.to_normed_group = normed_group.of_core F inner_product_space.of_core.to_normed_group._proof_1

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def inner_product_space.of_core.to_normed_space {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] [c : inner_product_space.core 𝕜 F] :

normed_space 𝕜 F

Normed space structure constructed from a inner_product_space.core structure

Equations

inner_product_space.of_core.to_normed_space = {to_module := _inst_3, norm_smul_le := _}

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def inner_product_space.of_core {𝕜 : Type u_1} {F : Type u_3} [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] (c : inner_product_space.core 𝕜 F) :

inner_product_space 𝕜 F

Given a inner_product_space.core structure on a space, one can use it to turn the space into an inner product space, constructing the norm out of the inner product

Equations

inner_product_space.of_core c = let _inst : normed_group F := inner_product_space.of_core.to_normed_group, _inst_4 : normed_space 𝕜 F := inner_product_space.of_core.to_normed_space in {to_normed_group := _inst, to_normed_space := _inst_4, to_has_inner := {inner := c.inner}, norm_sq_eq_inner := _, conj_sym := _, add_left := _, smul_left := _}

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@[simp]

theorem inner_conj_sym {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

⇑is_R_or_C.conj (inner y x) = inner x y

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theorem real_inner_comm {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

inner y x = inner x y

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theorem inner_eq_zero_sym {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner x y = 0 ↔ inner y x = 0

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@[simp]

theorem inner_self_nonneg_im {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

⇑is_R_or_C.im (inner x x) = 0

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theorem inner_self_im_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

⇑is_R_or_C.im (inner x x) = 0

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theorem inner_add_left {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y z : E} :

inner (x + y) z = inner x z + inner y z

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theorem inner_add_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y z : E} :

inner x (y + z) = inner x y + inner x z

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theorem inner_re_symm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

⇑is_R_or_C.re (inner x y) = ⇑is_R_or_C.re (inner y x)

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theorem inner_im_symm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

⇑is_R_or_C.im (inner x y) = -⇑is_R_or_C.im (inner y x)

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theorem inner_smul_left {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} {r : 𝕜} :

inner (r • x) y = (⇑is_R_or_C.conj r) * inner x y

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theorem real_inner_smul_left {F : Type u_3} [inner_product_space ℝ F] {x y : F} {r : ℝ} :

inner (r • x) y = r * inner x y

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theorem inner_smul_real_left {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} {r : ℝ} :

inner (↑r • x) y = r • inner x y

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theorem inner_smul_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} {r : 𝕜} :

inner x (r • y) = r * inner x y

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theorem real_inner_smul_right {F : Type u_3} [inner_product_space ℝ F] {x y : F} {r : ℝ} :

inner x (r • y) = r * inner x y

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theorem inner_smul_real_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} {r : ℝ} :

inner x (↑r • y) = r • inner x y

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def sesq_form_of_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

sesq_form 𝕜 E (is_R_or_C.conj_to_ring_equiv 𝕜)

The inner product as a sesquilinear form.

Equations

sesq_form_of_inner = {sesq := λ (x y : E), inner y x, sesq_add_left := _, sesq_smul_left := _, sesq_add_right := _, sesq_smul_right := _}

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@[simp]

theorem sesq_form_of_inner_sesq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

⇑sesq_form_of_inner x y = inner y x

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def bilin_form_of_real_inner {F : Type u_3} [inner_product_space ℝ F] :

bilin_form ℝ F

The real inner product as a bilinear form.

Equations

bilin_form_of_real_inner = {bilin := inner inner_product_space.to_has_inner, bilin_add_left := _, bilin_smul_left := _, bilin_add_right := _, bilin_smul_right := _}

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@[simp]

theorem bilin_form_of_real_inner_apply {F : Type u_3} [inner_product_space ℝ F] (ᾰ ᾰ_1 : F) :

⇑bilin_form_of_real_inner ᾰ ᾰ_1 = inner ᾰ ᾰ_1

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theorem sum_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_3} (s : finset ι) (f : ι → E) (x : E) :

inner (∑ (i : ι) in s, f i) x = ∑ (i : ι) in s, inner (f i) x

An inner product with a sum on the left.

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theorem inner_sum {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_3} (s : finset ι) (f : ι → E) (x : E) :

inner x (∑ (i : ι) in s, f i) = ∑ (i : ι) in s, inner x (f i)

An inner product with a sum on the right.

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theorem finsupp.sum_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_3} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :

inner (l.sum (λ (i : ι) (a : 𝕜), a • v i)) x = l.sum (λ (i : ι) (a : 𝕜), ⇑is_R_or_C.conj a • inner (v i) x)

An inner product with a sum on the left, finsupp version.

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theorem finsupp.inner_sum {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_3} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :

inner x (l.sum (λ (i : ι) (a : 𝕜), a • v i)) = l.sum (λ (i : ι) (a : 𝕜), a • inner x (v i))

An inner product with a sum on the right, finsupp version.

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@[simp]

theorem inner_zero_left {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

inner 0 x = 0

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theorem inner_re_zero_left {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

⇑is_R_or_C.re (inner 0 x) = 0

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@[simp]

theorem inner_zero_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

inner x 0 = 0

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theorem inner_re_zero_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

⇑is_R_or_C.re (inner x 0) = 0

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theorem inner_self_nonneg {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

0 ≤ ⇑is_R_or_C.re (inner x x)

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theorem real_inner_self_nonneg {F : Type u_3} [inner_product_space ℝ F] {x : F} :

0 ≤ inner x x

source

@[simp]

theorem inner_self_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

inner x x = 0 ↔ x = 0

source

@[simp]

theorem inner_self_nonpos {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

⇑is_R_or_C.re (inner x x) ≤ 0 ↔ x = 0

source

theorem real_inner_self_nonpos {F : Type u_3} [inner_product_space ℝ F] {x : F} :

inner x x ≤ 0 ↔ x = 0

source

@[simp]

theorem inner_self_re_to_K {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

↑(⇑is_R_or_C.re (inner x x)) = inner x x

source

theorem inner_self_eq_norm_sq_to_K {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x : E) :

inner x x = ↑∥x∥ ^ 2

source

theorem inner_self_re_abs {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

⇑is_R_or_C.re (inner x x) = is_R_or_C.abs (inner x x)

source

theorem inner_self_abs_to_K {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

↑(is_R_or_C.abs (inner x x)) = inner x x

source

theorem real_inner_self_abs {F : Type u_3} [inner_product_space ℝ F] {x : F} :

abs (inner x x) = inner x x

source

theorem inner_abs_conj_sym {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

is_R_or_C.abs (inner x y) = is_R_or_C.abs (inner y x)

source

@[simp]

theorem inner_neg_left {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner (-x) y = -inner x y

source

@[simp]

theorem inner_neg_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner x (-y) = -inner x y

source

theorem inner_neg_neg {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner (-x) (-y) = inner x y

source

@[simp]

theorem inner_self_conj {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} :

⇑is_R_or_C.conj (inner x x) = inner x x

source

theorem inner_sub_left {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y z : E} :

inner (x - y) z = inner x z - inner y z

source

theorem inner_sub_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y z : E} :

inner x (y - z) = inner x y - inner x z

source

theorem inner_mul_conj_re_abs {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

⇑is_R_or_C.re ((inner x y) * inner y x) = is_R_or_C.abs ((inner x y) * inner y x)

source

theorem inner_add_add_self {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner (x + y) (x + y) = inner x x + inner x y + inner y x + inner y y

Expand ⟪x + y, x + y⟫

source

theorem real_inner_add_add_self {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

inner (x + y) (x + y) = inner x x + 2 * inner x y + inner y y

Expand ⟪x + y, x + y⟫_ℝ

source

theorem inner_sub_sub_self {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner (x - y) (x - y) = inner x x - inner x y - inner y x + inner y y

source

theorem real_inner_sub_sub_self {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

inner (x - y) (x - y) = inner x x - 2 * inner x y + inner y y

Expand ⟪x - y, x - y⟫_ℝ

source

theorem parallelogram_law {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner (x + y) (x + y) + inner (x - y) (x - y) = 2 * (inner x x + inner y y)

Parallelogram law

source

theorem inner_mul_inner_self_le {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

(is_R_or_C.abs (inner x y)) * is_R_or_C.abs (inner y x) ≤ (⇑is_R_or_C.re (inner x x)) * ⇑is_R_or_C.re (inner y y)

Cauchy–Schwarz inequality. This proof follows "Proof 2" on Wikipedia.

source

theorem real_inner_mul_inner_self_le {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

(inner x y) * inner x y ≤ (inner x x) * inner y y

Cauchy–Schwarz inequality for real inner products.

source

theorem linear_independent_of_ne_zero_of_inner_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_3} {v : ι → E} (hz : ∀ (i : ι), v i ≠ 0) (ho : ∀ (i j : ι), i ≠ j → inner (v i) (v j) = 0) :

linear_independent 𝕜 v

A family of vectors is linearly independent if they are nonzero and orthogonal.

source

def orthonormal (𝕜 : Type u_1) {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} (v : ι → E) :

Prop

An orthonormal set of vectors in an inner_product_space

Equations

orthonormal 𝕜 v = ((∀ (i : ι), ∥v i∥ = 1) ∧ ∀ {i j : ι}, i ≠ j → inner (v i) (v j) = 0)

source

theorem orthonormal_iff_ite {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} {v : ι → E} :

orthonormal 𝕜 v ↔ ∀ (i j : ι), inner (v i) (v j) = ite (i = j) 1 0

if ... then ... else characterization of an indexed set of vectors being orthonormal. (Inner product equals Kronecker delta.)

source

theorem orthonormal_subtype_iff_ite {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {s : set E} :

orthonormal 𝕜 coe ↔ ∀ (v : E), v ∈ s → ∀ (w : E), w ∈ s → inner v w = ite (v = w) 1 0

if ... then ... else characterization of a set of vectors being orthonormal. (Inner product equals Kronecker delta.)

source

theorem orthonormal.inner_right_finsupp {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :

inner (v i) (⇑(finsupp.total ι E 𝕜 v) l) = ⇑l i

The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector.

source

theorem orthonormal.inner_right_fintype {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} [fintype ι] {v : ι → E} (hv : orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι) :

inner (v i) (∑ (i : ι), l i • v i) = l i

The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector.

source

theorem orthonormal.inner_left_finsupp {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :

inner (⇑(finsupp.total ι E 𝕜 v) l) (v i) = ⇑is_R_or_C.conj (⇑l i)

The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector.

source

theorem orthonormal.inner_left_fintype {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} [fintype ι] {v : ι → E} (hv : orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι) :

inner (∑ (i : ι), l i • v i) (v i) = ⇑is_R_or_C.conj (l i)

The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector.

source

theorem orthonormal.linear_independent {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : orthonormal 𝕜 v) :

linear_independent 𝕜 v

An orthonormal set is linearly independent.

source

theorem orthonormal.comp {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} {ι' : Type u_3} {v : ι → E} (hv : orthonormal 𝕜 v) (f : ι' → ι) (hf : function.injective f) :

orthonormal 𝕜 (v ∘ f)

A subfamily of an orthonormal family (i.e., a composition with an injective map) is an orthonormal family.

source

theorem orthonormal.inner_finsupp_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : orthonormal 𝕜 v) {s : set ι} {i : ι} (hi : i ∉ s) {l : ι →₀ 𝕜} (hl : l ∈ finsupp.supported 𝕜 𝕜 s) :

inner (⇑(finsupp.total ι E 𝕜 v) l) (v i) = 0

A linear combination of some subset of an orthonormal set is orthogonal to other members of the set.

source

theorem orthonormal_empty (𝕜 : Type u_1) (E : Type u_2) [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

orthonormal 𝕜 (λ (x : ↥∅), ↑x)

source

theorem orthonormal_Union_of_directed {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {η : Type u_3} {s : η → set E} (hs : directed has_subset.subset s) (h : ∀ (i : η), orthonormal 𝕜 (λ (x : ↥(s i)), ↑x)) :

orthonormal 𝕜 (λ (x : ↥⋃ (i : η), s i), ↑x)

source

theorem orthonormal_sUnion_of_directed {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {s : set (set E)} (hs : directed_on has_subset.subset s) (h : ∀ (a : set E), a ∈ s → orthonormal 𝕜 (λ (x : ↥a), ↑x)) :

orthonormal 𝕜 (λ (x : ↥⋃₀s), ↑x)

source

theorem exists_maximal_orthonormal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {s : set E} (hs : orthonormal 𝕜 coe) :

∃ (w : set E) (H : w ⊇ s), orthonormal 𝕜 coe ∧ ∀ (u : set E), u ⊇ w → orthonormal 𝕜 coe → u = w

Given an orthonormal set v of vectors in E, there exists a maximal orthonormal set containing it.

source

theorem orthonormal.ne_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : orthonormal 𝕜 v) (i : ι) :

v i ≠ 0

source

def basis_of_orthonormal_of_card_eq_finrank {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} [fintype ι] [nonempty ι] {v : ι → E} (hv : orthonormal 𝕜 v) (card_eq : fintype.card ι = finite_dimensional.finrank 𝕜 E) :

basis ι 𝕜 E

A family of orthonormal vectors with the correct cardinality forms a basis.

Equations

basis_of_orthonormal_of_card_eq_finrank hv card_eq = basis_of_linear_independent_of_card_eq_finrank _ card_eq

source

@[simp]

theorem coe_basis_of_orthonormal_of_card_eq_finrank {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} [fintype ι] [nonempty ι] {v : ι → E} (hv : orthonormal 𝕜 v) (card_eq : fintype.card ι = finite_dimensional.finrank 𝕜 E) :

⇑(basis_of_orthonormal_of_card_eq_finrank hv card_eq) = v

source

theorem norm_eq_sqrt_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x : E) :

∥x∥ = real.sqrt (⇑is_R_or_C.re (inner x x))

source

theorem norm_eq_sqrt_real_inner {F : Type u_3} [inner_product_space ℝ F] (x : F) :

∥x∥ = real.sqrt (inner x x)

source

theorem inner_self_eq_norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x : E) :

⇑is_R_or_C.re (inner x x) = ∥x∥ * ∥x∥

source

theorem real_inner_self_eq_norm_sq {F : Type u_3} [inner_product_space ℝ F] (x : F) :

inner x x = ∥x∥ * ∥x∥

source

theorem norm_add_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

∥x + y∥ ^ 2 = ∥x∥ ^ 2 + 2 * ⇑is_R_or_C.re (inner x y) + ∥y∥ ^ 2

Expand the square

source

theorem norm_add_pow_two {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

∥x + y∥ ^ 2 = ∥x∥ ^ 2 + 2 * ⇑is_R_or_C.re (inner x y) + ∥y∥ ^ 2

Alias of norm_add_sq.

source

theorem norm_add_sq_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

∥x + y∥ ^ 2 = ∥x∥ ^ 2 + 2 * inner x y + ∥y∥ ^ 2

Expand the square

source

theorem norm_add_pow_two_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

∥x + y∥ ^ 2 = ∥x∥ ^ 2 + 2 * inner x y + ∥y∥ ^ 2

Alias of norm_add_sq_real.

source

theorem norm_add_mul_self {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

∥x + y∥ * ∥x + y∥ = ∥x∥ * ∥x∥ + 2 * ⇑is_R_or_C.re (inner x y) + ∥y∥ * ∥y∥

Expand the square

source

theorem norm_add_mul_self_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

∥x + y∥ * ∥x + y∥ = ∥x∥ * ∥x∥ + 2 * inner x y + ∥y∥ * ∥y∥

Expand the square

source

theorem norm_sub_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

∥x - y∥ ^ 2 = ∥x∥ ^ 2 - 2 * ⇑is_R_or_C.re (inner x y) + ∥y∥ ^ 2

Expand the square

source

theorem norm_sub_pow_two {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

∥x - y∥ ^ 2 = ∥x∥ ^ 2 - 2 * ⇑is_R_or_C.re (inner x y) + ∥y∥ ^ 2

Alias of norm_sub_sq.

source

theorem norm_sub_sq_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

∥x - y∥ ^ 2 = ∥x∥ ^ 2 - 2 * inner x y + ∥y∥ ^ 2

Expand the square

source

theorem norm_sub_pow_two_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

∥x - y∥ ^ 2 = ∥x∥ ^ 2 - 2 * inner x y + ∥y∥ ^ 2

Alias of norm_sub_sq_real.

source

theorem norm_sub_mul_self {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

∥x - y∥ * ∥x - y∥ = ∥x∥ * ∥x∥ - 2 * ⇑is_R_or_C.re (inner x y) + ∥y∥ * ∥y∥

Expand the square

source

theorem norm_sub_mul_self_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

∥x - y∥ * ∥x - y∥ = ∥x∥ * ∥x∥ - 2 * inner x y + ∥y∥ * ∥y∥

Expand the square

source

theorem abs_inner_le_norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

is_R_or_C.abs (inner x y) ≤ ∥x∥ * ∥y∥

Cauchy–Schwarz inequality with norm

source

theorem abs_real_inner_le_norm {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

abs (inner x y) ≤ ∥x∥ * ∥y∥

Cauchy–Schwarz inequality with norm

source

theorem real_inner_le_norm {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

inner x y ≤ ∥x∥ * ∥y∥

Cauchy–Schwarz inequality with norm

source

theorem parallelogram_law_with_norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

∥x + y∥ * ∥x + y∥ + ∥x - y∥ * ∥x - y∥ = 2 * (∥x∥ * ∥x∥ + ∥y∥ * ∥y∥)

source

theorem parallelogram_law_with_norm_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

∥x + y∥ * ∥x + y∥ + ∥x - y∥ * ∥x - y∥ = 2 * (∥x∥ * ∥x∥ + ∥y∥ * ∥y∥)

source

theorem re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

⇑is_R_or_C.re (inner x y) = (∥x + y∥ * ∥x + y∥ - ∥x∥ * ∥x∥ - ∥y∥ * ∥y∥) / 2

Polarization identity: The real part of the inner product, in terms of the norm.

source

theorem re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

⇑is_R_or_C.re (inner x y) = (∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ - ∥x - y∥ * ∥x - y∥) / 2

Polarization identity: The real part of the inner product, in terms of the norm.

source

theorem re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

⇑is_R_or_C.re (inner x y) = (∥x + y∥ * ∥x + y∥ - ∥x - y∥ * ∥x - y∥) / 4

Polarization identity: The real part of the inner product, in terms of the norm.

source

theorem im_inner_eq_norm_sub_I_smul_mul_self_sub_norm_add_I_smul_mul_self_div_four {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

⇑is_R_or_C.im (inner x y) = (∥x - is_R_or_C.I • y∥ * ∥x - is_R_or_C.I • y∥ - ∥x + is_R_or_C.I • y∥ * ∥x + is_R_or_C.I • y∥) / 4

Polarization identity: The imaginary part of the inner product, in terms of the norm.

source

theorem inner_eq_sum_norm_sq_div_four {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

inner x y = (↑∥x + y∥ ^ 2 - ↑∥x - y∥ ^ 2 + (↑∥x - is_R_or_C.I • y∥ ^ 2 - ↑∥x + is_R_or_C.I • y∥ ^ 2) * is_R_or_C.I) / 4

Polarization identity: The inner product, in terms of the norm.

source

@[simp]

theorem linear_isometry.inner_map_map {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E →ₗᵢ[𝕜] E') (x y : E) :

inner (⇑f x) (⇑f y) = inner x y

A linear isometry preserves the inner product.

source

@[simp]

theorem linear_isometry_equiv.inner_map_map {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (x y : E) :

inner (⇑f x) (⇑f y) = inner x y

A linear isometric equivalence preserves the inner product.

source

def linear_map.isometry_of_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E →ₗ[𝕜] E') (h : ∀ (x y : E), inner (⇑f x) (⇑f y) = inner x y) :

E →ₗᵢ[𝕜] E'

A linear map that preserves the inner product is a linear isometry.

Equations

f.isometry_of_inner h = {to_linear_map := f, norm_map' := _}

source

@[simp]

theorem linear_map.coe_isometry_of_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E →ₗ[𝕜] E') (h : ∀ (x y : E), inner (⇑f x) (⇑f y) = inner x y) :

⇑(f.isometry_of_inner h) = ⇑f

source

@[simp]

theorem linear_map.isometry_of_inner_to_linear_map {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E →ₗ[𝕜] E') (h : ∀ (x y : E), inner (⇑f x) (⇑f y) = inner x y) :

(f.isometry_of_inner h).to_linear_map = f

source

def linear_equiv.isometry_of_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E ≃ₗ[𝕜] E') (h : ∀ (x y : E), inner (⇑f x) (⇑f y) = inner x y) :

E ≃ₗᵢ[𝕜] E'

A linear equivalence that preserves the inner product is a linear isometric equivalence.

Equations

f.isometry_of_inner h = {to_linear_equiv := f, norm_map' := _}

source

@[simp]

theorem linear_equiv.coe_isometry_of_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E ≃ₗ[𝕜] E') (h : ∀ (x y : E), inner (⇑f x) (⇑f y) = inner x y) :

⇑(f.isometry_of_inner h) = ⇑f

source

@[simp]

theorem linear_equiv.isometry_of_inner_to_linear_equiv {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {E' : Type u_4} [inner_product_space 𝕜 E'] (f : E ≃ₗ[𝕜] E') (h : ∀ (x y : E), inner (⇑f x) (⇑f y) = inner x y) :

(f.isometry_of_inner h).to_linear_equiv = f

source

theorem real_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

inner x y = (∥x + y∥ * ∥x + y∥ - ∥x∥ * ∥x∥ - ∥y∥ * ∥y∥) / 2

Polarization identity: The real inner product, in terms of the norm.

source

theorem real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

inner x y = (∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ - ∥x - y∥ * ∥x - y∥) / 2

Polarization identity: The real inner product, in terms of the norm.

source

theorem norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

∥x + y∥ * ∥x + y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ ↔ inner x y = 0

Pythagorean theorem, if-and-only-if vector inner product form.

source

theorem norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) (h : inner x y = 0) :

∥x + y∥ * ∥x + y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥

Pythagorean theorem, vector inner product form.

source

theorem norm_add_sq_eq_norm_sq_add_norm_sq_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} (h : inner x y = 0) :

∥x + y∥ * ∥x + y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥

Pythagorean theorem, vector inner product form.

source

theorem norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

∥x - y∥ * ∥x - y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥ ↔ inner x y = 0

Pythagorean theorem, subtracting vectors, if-and-only-if vector inner product form.

source

theorem norm_sub_sq_eq_norm_sq_add_norm_sq_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} (h : inner x y = 0) :

∥x - y∥ * ∥x - y∥ = ∥x∥ * ∥x∥ + ∥y∥ * ∥y∥

Pythagorean theorem, subtracting vectors, vector inner product form.

source

theorem real_inner_add_sub_eq_zero_iff {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

inner (x + y) (x - y) = 0 ↔ ∥x∥ = ∥y∥

The sum and difference of two vectors are orthogonal if and only if they have the same norm.

source

theorem abs_real_inner_div_norm_mul_norm_le_one {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

abs (inner x y / ∥x∥ * ∥y∥) ≤ 1

The real inner product of two vectors, divided by the product of their norms, has absolute value at most 1.

source

theorem real_inner_smul_self_left {F : Type u_3} [inner_product_space ℝ F] (x : F) (r : ℝ) :

inner (r • x) x = r * ∥x∥ * ∥x∥

The inner product of a vector with a multiple of itself.

source

theorem real_inner_smul_self_right {F : Type u_3} [inner_product_space ℝ F] (x : F) (r : ℝ) :

inner x (r • x) = r * ∥x∥ * ∥x∥

The inner product of a vector with a multiple of itself.

source

theorem abs_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x : E} {r : 𝕜} (hx : x ≠ 0) (hr : r ≠ 0) :

is_R_or_C.abs (inner x (r • x)) / ∥x∥ * ∥r • x∥ = 1

The inner product of a nonzero vector with a nonzero multiple of itself, divided by the product of their norms, has absolute value 1.

source

theorem abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul {F : Type u_3} [inner_product_space ℝ F] {x : F} {r : ℝ} (hx : x ≠ 0) (hr : r ≠ 0) :

abs (inner x (r • x)) / ∥x∥ * ∥r • x∥ = 1

The inner product of a nonzero vector with a nonzero multiple of itself, divided by the product of their norms, has absolute value 1.

source

theorem real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul {F : Type u_3} [inner_product_space ℝ F] {x : F} {r : ℝ} (hx : x ≠ 0) (hr : 0 < r) :

inner x (r • x) / ∥x∥ * ∥r • x∥ = 1

The inner product of a nonzero vector with a positive multiple of itself, divided by the product of their norms, has value 1.

source

theorem real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul {F : Type u_3} [inner_product_space ℝ F] {x : F} {r : ℝ} (hx : x ≠ 0) (hr : r < 0) :

inner x (r • x) / ∥x∥ * ∥r • x∥ = -1

The inner product of a nonzero vector with a negative multiple of itself, divided by the product of their norms, has value -1.

source

theorem abs_inner_div_norm_mul_norm_eq_one_iff {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

is_R_or_C.abs (inner x y / (↑∥x∥) * ↑∥y∥) = 1 ↔ x ≠ 0 ∧ ∃ (r : 𝕜), r ≠ 0 ∧ y = r • x

The inner product of two vectors, divided by the product of their norms, has absolute value 1 if and only if they are nonzero and one is a multiple of the other. One form of equality case for Cauchy-Schwarz.

source

theorem abs_real_inner_div_norm_mul_norm_eq_one_iff {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

abs (inner x y / ∥x∥ * ∥y∥) = 1 ↔ x ≠ 0 ∧ ∃ (r : ℝ), r ≠ 0 ∧ y = r • x

The inner product of two vectors, divided by the product of their norms, has absolute value 1 if and only if they are nonzero and one is a multiple of the other. One form of equality case for Cauchy-Schwarz.

source

theorem abs_inner_eq_norm_iff {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) (hx0 : x ≠ 0) (hy0 : y ≠ 0) :

is_R_or_C.abs (inner x y) = ∥x∥ * ∥y∥ ↔ ∃ (r : 𝕜), r ≠ 0 ∧ y = r • x

If the inner product of two vectors is equal to the product of their norms, then the two vectors are multiples of each other. One form of the equality case for Cauchy-Schwarz. Compare inner_eq_norm_mul_iff, which takes the stronger hypothesis ⟪x, y⟫ = ∥x∥ * ∥y∥.

source

theorem real_inner_div_norm_mul_norm_eq_one_iff {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

inner x y / ∥x∥ * ∥y∥ = 1 ↔ x ≠ 0 ∧ ∃ (r : ℝ), 0 < r ∧ y = r • x

The inner product of two vectors, divided by the product of their norms, has value 1 if and only if they are nonzero and one is a positive multiple of the other.

source

theorem real_inner_div_norm_mul_norm_eq_neg_one_iff {F : Type u_3} [inner_product_space ℝ F] (x y : F) :

inner x y / ∥x∥ * ∥y∥ = -1 ↔ x ≠ 0 ∧ ∃ (r : ℝ), r < 0 ∧ y = r • x

The inner product of two vectors, divided by the product of their norms, has value -1 if and only if they are nonzero and one is a negative multiple of the other.

source

theorem inner_eq_norm_mul_iff {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} :

inner x y = (↑∥x∥) * ↑∥y∥ ↔ ↑∥y∥ • x = ↑∥x∥ • y

If the inner product of two vectors is equal to the product of their norms (i.e., ⟪x, y⟫ = ∥x∥ * ∥y∥), then the two vectors are nonnegative real multiples of each other. One form of the equality case for Cauchy-Schwarz. Compare abs_inner_eq_norm_iff, which takes the weaker hypothesis abs ⟪x, y⟫ = ∥x∥ * ∥y∥.

source

theorem inner_eq_norm_mul_iff_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

inner x y = ∥x∥ * ∥y∥ ↔ ∥y∥ • x = ∥x∥ • y

If the inner product of two vectors is equal to the product of their norms (i.e., ⟪x, y⟫ = ∥x∥ * ∥y∥), then the two vectors are nonnegative real multiples of each other. One form of the equality case for Cauchy-Schwarz. Compare abs_inner_eq_norm_iff, which takes the weaker hypothesis abs ⟪x, y⟫ = ∥x∥ * ∥y∥.

source

theorem inner_eq_norm_mul_iff_of_norm_one {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {x y : E} (hx : ∥x∥ = 1) (hy : ∥y∥ = 1) :

inner x y = 1 ↔ x = y

If the inner product of two unit vectors is 1, then the two vectors are equal. One form of the equality case for Cauchy-Schwarz.

source

theorem inner_lt_norm_mul_iff_real {F : Type u_3} [inner_product_space ℝ F] {x y : F} :

inner x y < ∥x∥ * ∥y∥ ↔ ∥y∥ • x ≠ ∥x∥ • y

source

theorem inner_lt_one_iff_real_of_norm_one {F : Type u_3} [inner_product_space ℝ F] {x y : F} (hx : ∥x∥ = 1) (hy : ∥y∥ = 1) :

inner x y < 1 ↔ x ≠ y

If the inner product of two unit vectors is strictly less than 1, then the two vectors are distinct. One form of the equality case for Cauchy-Schwarz.

source

theorem inner_sum_smul_sum_smul_of_sum_eq_zero {F : Type u_3} [inner_product_space ℝ F] {ι₁ : Type u_1} {s₁ : finset ι₁} {w₁ : ι₁ → ℝ} (v₁ : ι₁ → F) (h₁ : ∑ (i : ι₁) in s₁, w₁ i = 0) {ι₂ : Type u_2} {s₂ : finset ι₂} {w₂ : ι₂ → ℝ} (v₂ : ι₂ → F) (h₂ : ∑ (i : ι₂) in s₂, w₂ i = 0) :

inner (∑ (i₁ : ι₁) in s₁, w₁ i₁ • v₁ i₁) (∑ (i₂ : ι₂) in s₂, w₂ i₂ • v₂ i₂) = (-∑ (i₁ : ι₁) in s₁, ∑ (i₂ : ι₂) in s₂, ((w₁ i₁) * w₂ i₂) * ∥v₁ i₁ - v₂ i₂∥ * ∥v₁ i₁ - v₂ i₂∥) / 2

The inner product of two weighted sums, where the weights in each sum add to 0, in terms of the norms of pairwise differences.

source

def inner_right {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (v : E) :

E →L[𝕜] 𝕜

The inner product with a fixed left element, as a continuous linear map. This can be upgraded to a continuous map which is jointly conjugate-linear in the left argument and linear in the right argument, once (TODO) conjugate-linear maps have been defined.

Equations

inner_right v = {to_fun := λ (w : E), inner v w, map_add' := _, map_smul' := _}.mk_continuous ∥v∥ _

source

@[simp]

theorem inner_right_coe {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (v : E) :

⇑(inner_right v) = λ (w : E), inner v w

source

@[simp]

theorem inner_right_apply {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (v w : E) :

⇑(inner_right v) w = inner v w

source

@[protected, instance]

def pi_Lp.inner_product_space {𝕜 : Type u_1} [is_R_or_C 𝕜] {ι : Type u_2} [fintype ι] (f : ι → Type u_3) [Π (i : ι), inner_product_space 𝕜 (f i)] :

inner_product_space 𝕜 (pi_Lp 2 one_le_two f)

Equations

pi_Lp.inner_product_space f = {to_normed_group := pi_Lp.normed_group 2 one_le_two f (λ (i : ι), inner_product_space.to_normed_group 𝕜), to_normed_space := pi_Lp.normed_space 2 one_le_two f 𝕜 (λ (i : ι), inner_product_space.to_normed_space), to_has_inner := {inner := λ (x y : pi_Lp 2 one_le_two f), ∑ (i : ι), inner (x i) (y i)}, norm_sq_eq_inner := _, conj_sym := _, add_left := _, smul_left := _}

source

@[simp]

theorem pi_Lp.inner_apply {𝕜 : Type u_1} [is_R_or_C 𝕜] {ι : Type u_2} [fintype ι] {f : ι → Type u_3} [Π (i : ι), inner_product_space 𝕜 (f i)] (x y : pi_Lp 2 one_le_two f) :

inner x y = ∑ (i : ι), inner (x i) (y i)

source

theorem pi_Lp.norm_eq_of_L2 {𝕜 : Type u_1} [is_R_or_C 𝕜] {ι : Type u_2} [fintype ι] {f : ι → Type u_3} [Π (i : ι), inner_product_space 𝕜 (f i)] (x : pi_Lp 2 one_le_two f) :

∥x∥ = real.sqrt (∑ (i : ι), ∥x i∥ ^ 2)

source

@[protected, instance]

def is_R_or_C.inner_product_space {𝕜 : Type u_1} [is_R_or_C 𝕜] :

inner_product_space 𝕜 𝕜

A field 𝕜 satisfying is_R_or_C is itself a 𝕜-inner product space.

Equations

is_R_or_C.inner_product_space = {to_normed_group := normed_ring.to_normed_group normed_comm_ring.to_normed_ring, to_normed_space := normed_field.to_normed_space nondiscrete_normed_field.to_normed_field, to_has_inner := {inner := λ (x y : 𝕜), (⇑is_R_or_C.conj x) * y}, norm_sq_eq_inner := _, conj_sym := _, add_left := _, smul_left := _}

source

@[simp]

theorem is_R_or_C.inner_apply {𝕜 : Type u_1} [is_R_or_C 𝕜] (x y : 𝕜) :

inner x y = (⇑is_R_or_C.conj x) * y

source

@[nolint]

def euclidean_space (𝕜 : Type u_1) [is_R_or_C 𝕜] (n : Type u_2) [fintype n] :

Type (max u_2 u_1)

The standard real/complex Euclidean space, functions on a finite type. For an n-dimensional space use euclidean_space 𝕜 (fin n).

Equations

euclidean_space 𝕜 n = pi_Lp 2 one_le_two (λ (i : n), 𝕜)

source

theorem euclidean_space.norm_eq {𝕜 : Type u_1} [is_R_or_C 𝕜] {n : Type u_2} [fintype n] (x : euclidean_space 𝕜 n) :

∥x∥ = real.sqrt (∑ (i : n), ∥x i∥ ^ 2)

source

@[protected, instance]

def submodule.inner_product_space {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (W : submodule 𝕜 E) :

inner_product_space 𝕜 ↥W

Induced inner product on a submodule.

Equations

W.inner_product_space = {to_normed_group := W.normed_group, to_normed_space := {to_module := normed_space.to_module W.normed_space, norm_smul_le := _}, to_has_inner := {inner := λ (x y : ↥W), inner ↑x ↑y}, norm_sq_eq_inner := _, conj_sym := _, add_left := _, smul_left := _}

source

@[simp]

theorem submodule.coe_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (W : submodule 𝕜 E) (x y : ↥W) :

inner x y = inner ↑x ↑y

The inner product on submodules is the same as on the ambient space.

source

def has_inner.is_R_or_C_to_real (𝕜 : Type u_1) (E : Type u_2) [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

has_inner ℝ E

A general inner product implies a real inner product. This is not registered as an instance since it creates problems with the case 𝕜 = ℝ.

Equations

has_inner.is_R_or_C_to_real 𝕜 E = {inner := λ (x y : E), ⇑is_R_or_C.re (inner x y)}

source

def inner_product_space.is_R_or_C_to_real (𝕜 : Type u_1) (E : Type u_2) [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

inner_product_space ℝ E

A general inner product space structure implies a real inner product structure. This is not registered as an instance since it creates problems with the case 𝕜 = ℝ, but in can be used in a proof to obtain a real inner product space structure from a given 𝕜-inner product space structure.

Equations

inner_product_space.is_R_or_C_to_real 𝕜 E = {to_normed_group := inner_product_space.to_normed_group 𝕜 _inst_2, to_normed_space := {to_module := normed_space.to_module (normed_space.restrict_scalars ℝ 𝕜 E), norm_smul_le := _}, to_has_inner := {inner := inner (has_inner.is_R_or_C_to_real 𝕜 E)}, norm_sq_eq_inner := _, conj_sym := _, add_left := _, smul_left := _}

source

theorem real_inner_eq_re_inner (𝕜 : Type u_1) {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

inner x y = ⇑is_R_or_C.re (inner x y)

source

@[protected, instance]

def inner_product_space.complex_to_real {G : Type u_4} [inner_product_space ℂ G] :

inner_product_space ℝ G

A complex inner product implies a real inner product

Equations

inner_product_space.complex_to_real = inner_product_space.is_R_or_C_to_real ℂ G

source

theorem is_bounded_bilinear_map_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] :

is_bounded_bilinear_map ℝ (λ (p : E × E), inner p.fst p.snd)

source

def fderiv_inner_clm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] (p : E × E) :

E × E →L[ℝ ] 𝕜

Derivative of the inner product.

Equations

fderiv_inner_clm p = is_bounded_bilinear_map_inner.deriv p

source

@[simp]

theorem fderiv_inner_clm_apply {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] (p x : E × E) :

⇑(fderiv_inner_clm p) x = inner p.fst x.snd + inner x.fst p.snd

source

theorem times_cont_diff_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {n : with_top ℕ} :

times_cont_diff ℝ n (λ (p : E × E), inner p.fst p.snd)

source

theorem times_cont_diff_at_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {p : E × E} {n : with_top ℕ} :

times_cont_diff_at ℝ n (λ (p : E × E), inner p.fst p.snd) p

source

theorem differentiable_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] :

differentiable ℝ (λ (p : E × E), inner p.fst p.snd)

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theorem times_cont_diff_within_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} {x : G} {n : with_top ℕ} (hf : times_cont_diff_within_at ℝ n f s x) (hg : times_cont_diff_within_at ℝ n g s x) :

times_cont_diff_within_at ℝ n (λ (x : G), inner (f x) (g x)) s x

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theorem times_cont_diff_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {x : G} {n : with_top ℕ} (hf : times_cont_diff_at ℝ n f x) (hg : times_cont_diff_at ℝ n g x) :

times_cont_diff_at ℝ n (λ (x : G), inner (f x) (g x)) x

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theorem times_cont_diff_on.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} {n : with_top ℕ} (hf : times_cont_diff_on ℝ n f s) (hg : times_cont_diff_on ℝ n g s) :

times_cont_diff_on ℝ n (λ (x : G), inner (f x) (g x)) s

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theorem times_cont_diff.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {n : with_top ℕ} (hf : times_cont_diff ℝ n f) (hg : times_cont_diff ℝ n g) :

times_cont_diff ℝ n (λ (x : G), inner (f x) (g x))

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theorem has_fderiv_within_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {f' g' : G →L[ℝ ] E} {s : set G} {x : G} (hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at g g' s x) :

has_fderiv_within_at (λ (t : G), inner (f t) (g t)) ((fderiv_inner_clm (f x, g x)).comp (f'.prod g')) s x

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theorem has_fderiv_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {f' g' : G →L[ℝ ] E} {x : G} (hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :

has_fderiv_at (λ (t : G), inner (f t) (g t)) ((fderiv_inner_clm (f x, g x)).comp (f'.prod g')) x

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theorem has_deriv_within_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {f g : ℝ → E} {f' g' : E} {s : set ℝ} {x : ℝ} (hf : has_deriv_within_at f f' s x) (hg : has_deriv_within_at g g' s x) :

has_deriv_within_at (λ (t : ℝ), inner (f t) (g t)) (inner (f x) g' + inner f' (g x)) s x

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theorem has_deriv_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {f g : ℝ → E} {f' g' : E} {x : ℝ} :

has_deriv_at f f' x → has_deriv_at g g' x → has_deriv_at (λ (t : ℝ), inner (f t) (g t)) (inner (f x) g' + inner f' (g x)) x

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theorem differentiable_within_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} {x : G} (hf : differentiable_within_at ℝ f s x) (hg : differentiable_within_at ℝ g s x) :

differentiable_within_at ℝ (λ (x : G), inner (f x) (g x)) s x

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theorem differentiable_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {x : G} (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x) :

differentiable_at ℝ (λ (x : G), inner (f x) (g x)) x

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theorem differentiable_on.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} (hf : differentiable_on ℝ f s) (hg : differentiable_on ℝ g s) :

differentiable_on ℝ (λ (x : G), inner (f x) (g x)) s

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theorem differentiable.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} (hf : differentiable ℝ f) (hg : differentiable ℝ g) :

differentiable ℝ (λ (x : G), inner (f x) (g x))

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theorem fderiv_inner_apply {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {x : G} (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x) (y : G) :

⇑(fderiv ℝ (λ (t : G), inner (f t) (g t)) x) y = inner (f x) (⇑(fderiv ℝ g x) y) + inner (⇑(fderiv ℝ f x) y) (g x)

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theorem deriv_inner_apply {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {f g : ℝ → E} {x : ℝ} (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x) :

deriv (λ (t : ℝ), inner (f t) (g t)) x = inner (f x) (deriv g x) + inner (deriv f x) (g x)

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theorem times_cont_diff_norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {n : with_top ℕ} :

times_cont_diff ℝ n (λ (x : E), ∥x∥ ^ 2)

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theorem times_cont_diff.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {n : with_top ℕ} (hf : times_cont_diff ℝ n f) :

times_cont_diff ℝ n (λ (x : G), ∥f x∥ ^ 2)

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theorem times_cont_diff_within_at.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} {x : G} {n : with_top ℕ} (hf : times_cont_diff_within_at ℝ n f s x) :

times_cont_diff_within_at ℝ n (λ (y : G), ∥f y∥ ^ 2) s x

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theorem times_cont_diff_at.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {x : G} {n : with_top ℕ} (hf : times_cont_diff_at ℝ n f x) :

times_cont_diff_at ℝ n (λ (y : G), ∥f y∥ ^ 2) x

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theorem times_cont_diff_at_norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {n : with_top ℕ} {x : E} (hx : x ≠ 0) :

times_cont_diff_at ℝ n norm x

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theorem times_cont_diff_at.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {x : G} {n : with_top ℕ} (hf : times_cont_diff_at ℝ n f x) (h0 : f x ≠ 0) :

times_cont_diff_at ℝ n (λ (y : G), ∥f y∥) x

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theorem times_cont_diff_at.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {x : G} {n : with_top ℕ} (hf : times_cont_diff_at ℝ n f x) (hg : times_cont_diff_at ℝ n g x) (hne : f x ≠ g x) :

times_cont_diff_at ℝ n (λ (y : G), dist (f y) (g y)) x

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theorem times_cont_diff_within_at.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} {x : G} {n : with_top ℕ} (hf : times_cont_diff_within_at ℝ n f s x) (h0 : f x ≠ 0) :

times_cont_diff_within_at ℝ n (λ (y : G), ∥f y∥) s x

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theorem times_cont_diff_within_at.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} {x : G} {n : with_top ℕ} (hf : times_cont_diff_within_at ℝ n f s x) (hg : times_cont_diff_within_at ℝ n g s x) (hne : f x ≠ g x) :

times_cont_diff_within_at ℝ n (λ (y : G), dist (f y) (g y)) s x

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theorem times_cont_diff_on.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} {n : with_top ℕ} (hf : times_cont_diff_on ℝ n f s) :

times_cont_diff_on ℝ n (λ (y : G), ∥f y∥ ^ 2) s

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theorem times_cont_diff_on.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} {n : with_top ℕ} (hf : times_cont_diff_on ℝ n f s) (h0 : ∀ (x : G), x ∈ s → f x ≠ 0) :

times_cont_diff_on ℝ n (λ (y : G), ∥f y∥) s

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theorem times_cont_diff_on.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} {n : with_top ℕ} (hf : times_cont_diff_on ℝ n f s) (hg : times_cont_diff_on ℝ n g s) (hne : ∀ (x : G), x ∈ s → f x ≠ g x) :

times_cont_diff_on ℝ n (λ (y : G), dist (f y) (g y)) s

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theorem times_cont_diff.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {n : with_top ℕ} (hf : times_cont_diff ℝ n f) (h0 : ∀ (x : G), f x ≠ 0) :

times_cont_diff ℝ n (λ (y : G), ∥f y∥)

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theorem times_cont_diff.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {n : with_top ℕ} (hf : times_cont_diff ℝ n f) (hg : times_cont_diff ℝ n g) (hne : ∀ (x : G), f x ≠ g x) :

times_cont_diff ℝ n (λ (y : G), dist (f y) (g y))

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theorem differentiable_at.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {x : G} (hf : differentiable_at ℝ f x) :

differentiable_at ℝ (λ (y : G), ∥f y∥ ^ 2) x

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theorem differentiable_at.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {x : G} (hf : differentiable_at ℝ f x) (h0 : f x ≠ 0) :

differentiable_at ℝ (λ (y : G), ∥f y∥) x

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theorem differentiable_at.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {x : G} (hf : differentiable_at ℝ f x) (hg : differentiable_at ℝ g x) (hne : f x ≠ g x) :

differentiable_at ℝ (λ (y : G), dist (f y) (g y)) x

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theorem differentiable.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} (hf : differentiable ℝ f) :

differentiable ℝ (λ (y : G), ∥f y∥ ^ 2)

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theorem differentiable.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} (hf : differentiable ℝ f) (h0 : ∀ (x : G), f x ≠ 0) :

differentiable ℝ (λ (y : G), ∥f y∥)

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theorem differentiable.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} (hf : differentiable ℝ f) (hg : differentiable ℝ g) (hne : ∀ (x : G), f x ≠ g x) :

differentiable ℝ (λ (y : G), dist (f y) (g y))

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theorem differentiable_within_at.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} {x : G} (hf : differentiable_within_at ℝ f s x) :

differentiable_within_at ℝ (λ (y : G), ∥f y∥ ^ 2) s x

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theorem differentiable_within_at.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} {x : G} (hf : differentiable_within_at ℝ f s x) (h0 : f x ≠ 0) :

differentiable_within_at ℝ (λ (y : G), ∥f y∥) s x

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theorem differentiable_within_at.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} {x : G} (hf : differentiable_within_at ℝ f s x) (hg : differentiable_within_at ℝ g s x) (hne : f x ≠ g x) :

differentiable_within_at ℝ (λ (y : G), dist (f y) (g y)) s x

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theorem differentiable_on.norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} (hf : differentiable_on ℝ f s) :

differentiable_on ℝ (λ (y : G), ∥f y∥ ^ 2) s

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theorem differentiable_on.norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f : G → E} {s : set G} (hf : differentiable_on ℝ f s) (h0 : ∀ (x : G), x ∈ s → f x ≠ 0) :

differentiable_on ℝ (λ (y : G), ∥f y∥) s

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theorem differentiable_on.dist {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] {G : Type u_4} [normed_group G] [normed_space ℝ G] {f g : G → E} {s : set G} (hf : differentiable_on ℝ f s) (hg : differentiable_on ℝ g s) (hne : ∀ (x : G), x ∈ s → f x ≠ g x) :

differentiable_on ℝ (λ (y : G), dist (f y) (g y)) s

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theorem continuous_inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

continuous (λ (p : E × E), inner p.fst p.snd)

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theorem filter.tendsto.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {α : Type u_4} {f g : α → E} {l : filter α} {x y : E} (hf : filter.tendsto f l (𝓝 x)) (hg : filter.tendsto g l (𝓝 y)) :

filter.tendsto (λ (t : α), inner (f t) (g t)) l (𝓝 (inner x y))

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theorem measurable.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {α : Type u_4} [measurable_space α] [measurable_space E] [opens_measurable_space E] [topological_space.second_countable_topology E] [measurable_space 𝕜] [borel_space 𝕜] {f g : α → E} (hf : measurable f) (hg : measurable g) :

measurable (λ (t : α), inner (f t) (g t))

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theorem ae_measurable.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {α : Type u_4} [measurable_space α] [measurable_space E] [opens_measurable_space E] [topological_space.second_countable_topology E] [measurable_space 𝕜] [borel_space 𝕜] {μ : measure_theory.measure α} {f g : α → E} (hf : ae_measurable f μ) (hg : ae_measurable g μ) :

ae_measurable (λ (x : α), inner (f x) (g x)) μ

source

theorem continuous_within_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {α : Type u_4} [topological_space α] {f g : α → E} {x : α} {s : set α} (hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :

continuous_within_at (λ (t : α), inner (f t) (g t)) s x

source

theorem continuous_at.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {α : Type u_4} [topological_space α] {f g : α → E} {x : α} (hf : continuous_at f x) (hg : continuous_at g x) :

continuous_at (λ (t : α), inner (f t) (g t)) x

source

theorem continuous_on.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {α : Type u_4} [topological_space α] {f g : α → E} {s : set α} (hf : continuous_on f s) (hg : continuous_on g s) :

continuous_on (λ (t : α), inner (f t) (g t)) s

source

theorem continuous.inner {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {α : Type u_4} [topological_space α] {f g : α → E} (hf : continuous f) (hg : continuous g) :

continuous (λ (t : α), inner (f t) (g t))

source

@[protected, instance]

def euclidean_space.finite_dimensional {𝕜 : Type u_1} [is_R_or_C 𝕜] {ι : Type u_4} [fintype ι] :

finite_dimensional 𝕜 (euclidean_space 𝕜 ι)

source

@[protected, instance]

def euclidean_space.inner_product_space {𝕜 : Type u_1} [is_R_or_C 𝕜] {ι : Type u_4} [fintype ι] :

inner_product_space 𝕜 (euclidean_space 𝕜 ι)

Equations

euclidean_space.inner_product_space = pi_Lp.inner_product_space (λ (i : ι), 𝕜)

source

@[simp]

theorem finrank_euclidean_space {𝕜 : Type u_1} [is_R_or_C 𝕜] {ι : Type u_4} [fintype ι] :

finite_dimensional.finrank 𝕜 (euclidean_space 𝕜 ι) = fintype.card ι

source

theorem finrank_euclidean_space_fin {𝕜 : Type u_1} [is_R_or_C 𝕜] {n : ℕ} :

finite_dimensional.finrank 𝕜 (euclidean_space 𝕜 (fin n)) = n

source

def basis.isometry_euclidean_of_orthonormal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_4} [fintype ι] (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 ⇑v) :

E ≃ₗᵢ[𝕜] euclidean_space 𝕜 ι

An orthonormal basis on a fintype ι for an inner product space induces an isometry with euclidean_space 𝕜 ι.

Equations

v.isometry_euclidean_of_orthonormal hv = v.equiv_fun.isometry_of_inner _

source

def complex.isometry_euclidean :

ℂ ≃ₗᵢ[ℝ ] euclidean_space ℝ (fin 2)

ℂ is isometric to ℝ² with the Euclidean inner product.

Equations

complex.isometry_euclidean = complex.basis_one_I.isometry_euclidean_of_orthonormal complex.isometry_euclidean._proof_1

source

@[simp]

theorem complex.isometry_euclidean_symm_apply (x : euclidean_space ℝ (fin 2)) :

⇑(complex.isometry_euclidean.symm) x = ↑(x 0) + (↑(x 1)) * is_R_or_C.I

source

theorem complex.isometry_euclidean_proj_eq_self (z : ℂ) :

↑(⇑complex.isometry_euclidean z 0) + (↑(⇑complex.isometry_euclidean z 1)) * is_R_or_C.I = z

source

@[simp]

theorem complex.isometry_euclidean_apply_zero (z : ℂ) :

⇑complex.isometry_euclidean z 0 = z.re

source

@[simp]

theorem complex.isometry_euclidean_apply_one (z : ℂ) :

⇑complex.isometry_euclidean z 1 = z.im

source

theorem exists_norm_eq_infi_of_complete_convex {F : Type u_3} [inner_product_space ℝ F] {K : set F} (ne : K.nonempty) (h₁ : is_complete K) (h₂ : convex K) (u : F) :

∃ (v : F) (H : v ∈ K), ∥u - v∥ = ⨅ (w : ↥K), ∥u - ↑w∥

Existence of minimizers Let u be a point in a real inner product space, and let K be a nonempty complete convex subset. Then there exists a (unique) v in K that minimizes the distance ∥u - v∥ to u.

source

theorem norm_eq_infi_iff_real_inner_le_zero {F : Type u_3} [inner_product_space ℝ F] {K : set F} (h : convex K) {u v : F} (hv : v ∈ K) :

(∥u - v∥ = ⨅ (w : ↥K), ∥u - ↑w∥) ↔ ∀ (w : F), w ∈ K → inner (u - v) (w - v) ≤ 0

Characterization of minimizers for the projection on a convex set in a real inner product space.

source

theorem exists_norm_eq_infi_of_complete_subspace {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) (h : is_complete ↑K) (u : E) :

∃ (v : E) (H : v ∈ K), ∥u - v∥ = ⨅ (w : ↥↑K), ∥u - ↑w∥

Existence of projections on complete subspaces. Let u be a point in an inner product space, and let K be a nonempty complete subspace. Then there exists a (unique) v in K that minimizes the distance ∥u - v∥ to u. This point v is usually called the orthogonal projection of u onto K.

source

theorem norm_eq_infi_iff_real_inner_eq_zero {F : Type u_3} [inner_product_space ℝ F] (K : submodule ℝ F) {u v : F} (hv : v ∈ K) :

(∥u - v∥ = ⨅ (w : ↥↑K), ∥u - ↑w∥) ↔ ∀ (w : F), w ∈ K → inner (u - v) w = 0

Characterization of minimizers in the projection on a subspace, in the real case. Let u be a point in a real inner product space, and let K be a nonempty subspace. Then point v minimizes the distance ∥u - v∥ over points in K if and only if for all w ∈ K, ⟪u - v, w⟫ = 0 (i.e., u - v is orthogonal to the subspace K). This is superceded by norm_eq_infi_iff_inner_eq_zero that gives the same conclusion over any is_R_or_C field.

source

theorem norm_eq_infi_iff_inner_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) {u v : E} (hv : v ∈ K) :

(∥u - v∥ = ⨅ (w : ↥↑K), ∥u - ↑w∥) ↔ ∀ (w : E), w ∈ K → inner (u - v) w = 0

Characterization of minimizers in the projection on a subspace. Let u be a point in an inner product space, and let K be a nonempty subspace. Then point v minimizes the distance ∥u - v∥ over points in K if and only if for all w ∈ K, ⟪u - v, w⟫ = 0 (i.e., u - v is orthogonal to the subspace K)

source

def orthogonal_projection_fn {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space ↥K] (v : E) :

E

The orthogonal projection onto a complete subspace, as an unbundled function. This definition is only intended for use in setting up the bundled version orthogonal_projection and should not be used once that is defined.

Equations

orthogonal_projection_fn K v = _.some

source

theorem orthogonal_projection_fn_mem {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] (v : E) :

orthogonal_projection_fn K v ∈ K

The unbundled orthogonal projection is in the given subspace. This lemma is only intended for use in setting up the bundled version and should not be used once that is defined.

source

theorem orthogonal_projection_fn_inner_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] (v w : E) (H : w ∈ K) :

inner (v - orthogonal_projection_fn K v) w = 0

The characterization of the unbundled orthogonal projection. This lemma is only intended for use in setting up the bundled version and should not be used once that is defined.

source

theorem eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] {u v : E} (hvm : v ∈ K) (hvo : ∀ (w : E), w ∈ K → inner (u - v) w = 0) :

orthogonal_projection_fn K u = v

The unbundled orthogonal projection is the unique point in K with the orthogonality property. This lemma is only intended for use in setting up the bundled version and should not be used once that is defined.

source

theorem orthogonal_projection_fn_norm_sq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space ↥K] (v : E) :

∥v∥ * ∥v∥ = ∥v - orthogonal_projection_fn K v∥ * ∥v - orthogonal_projection_fn K v∥ + ∥orthogonal_projection_fn K v∥ * ∥orthogonal_projection_fn K v∥

source

def orthogonal_projection {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space ↥K] :

E →L[𝕜] ↥K

The orthogonal projection onto a complete subspace.

Equations

orthogonal_projection K = {to_fun := λ (v : E), ⟨orthogonal_projection_fn K v, _⟩, map_add' := _, map_smul' := _}.mk_continuous 1 _

source

@[simp]

theorem orthogonal_projection_fn_eq {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] (v : E) :

orthogonal_projection_fn K v = ↑(⇑(orthogonal_projection K) v)

source

@[simp]

theorem orthogonal_projection_inner_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] (v w : E) (H : w ∈ K) :

inner (v - ↑(⇑(orthogonal_projection K) v)) w = 0

The characterization of the orthogonal projection.

source

theorem eq_orthogonal_projection_of_mem_of_inner_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] {u v : E} (hvm : v ∈ K) (hvo : ∀ (w : E), w ∈ K → inner (u - v) w = 0) :

↑(⇑(orthogonal_projection K) u) = v

The orthogonal projection is the unique point in K with the orthogonality property.

source

theorem eq_orthogonal_projection_of_eq_submodule {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] {K' : submodule 𝕜 E} [complete_space ↥K'] (h : K = K') (u : E) :

↑(⇑(orthogonal_projection K) u) = ↑(⇑(orthogonal_projection K') u)

The orthogonal projections onto equal subspaces are coerced back to the same point in E.

source

@[simp]

theorem orthogonal_projection_mem_subspace_eq_self {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] (v : ↥K) :

⇑(orthogonal_projection K) ↑v = v

The orthogonal projection sends elements of K to themselves.

source

@[simp]

theorem orthogonal_projection_bot {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

orthogonal_projection ⊥ = 0

The orthogonal projection onto the trivial submodule is the zero map.

source

theorem orthogonal_projection_norm_le {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space ↥K] :

∥orthogonal_projection K∥ ≤ 1

The orthogonal projection has norm ≤ 1.

source

theorem smul_orthogonal_projection_singleton (𝕜 : Type u_1) {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {v : E} (w : E) :

↑∥v∥ ^ 2 • ↑(⇑(orthogonal_projection (submodule.span 𝕜 {v})) w) = inner v w • v

source

theorem orthogonal_projection_singleton (𝕜 : Type u_1) {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {v : E} (w : E) :

↑(⇑(orthogonal_projection (submodule.span 𝕜 {v})) w) = (inner v w / ↑∥v∥ ^ 2) • v

Formula for orthogonal projection onto a single vector.

source

theorem orthogonal_projection_unit_singleton (𝕜 : Type u_1) {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {v : E} (hv : ∥v∥ = 1) (w : E) :

↑(⇑(orthogonal_projection (submodule.span 𝕜 {v})) w) = inner v w • v

Formula for orthogonal projection onto a single unit vector.

source

def submodule.orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) :

submodule 𝕜 E

The subspace of vectors orthogonal to a given subspace.

Equations

Kᗮ = {carrier := {v : E | ∀ (u : E), u ∈ K → inner u v = 0}, zero_mem' := _, add_mem' := _, smul_mem' := _}

source

theorem submodule.mem_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) (v : E) :

v ∈ Kᗮ ↔ ∀ (u : E), u ∈ K → inner u v = 0

When a vector is in Kᗮ.

source

theorem submodule.mem_orthogonal' {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) (v : E) :

v ∈ Kᗮ ↔ ∀ (u : E), u ∈ K → inner v u = 0

When a vector is in Kᗮ, with the inner product the other way round.

source

theorem submodule.inner_right_of_mem_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} {u v : E} (hu : u ∈ K) (hv : v ∈ Kᗮ) :

inner u v = 0

A vector in K is orthogonal to one in Kᗮ.

source

theorem submodule.inner_left_of_mem_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} {u v : E} (hu : u ∈ K) (hv : v ∈ Kᗮ) :

inner v u = 0

A vector in Kᗮ is orthogonal to one in K.

source

theorem inner_right_of_mem_orthogonal_singleton {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (u : E) {v : E} (hv : v ∈ (submodule.span 𝕜 {u})ᗮ) :

inner u v = 0

A vector in (𝕜 ∙ u)ᗮ is orthogonal to u.

source

theorem inner_left_of_mem_orthogonal_singleton {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (u : E) {v : E} (hv : v ∈ (submodule.span 𝕜 {u})ᗮ) :

inner v u = 0

A vector in (𝕜 ∙ u)ᗮ is orthogonal to u.

source

theorem submodule.inf_orthogonal_eq_bot {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) :

K ⊓ Kᗮ = ⊥

K and Kᗮ have trivial intersection.

source

theorem submodule.orthogonal_disjoint {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) :

disjoint K Kᗮ

K and Kᗮ have trivial intersection.

source

theorem orthogonal_eq_inter {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) :

Kᗮ = ⨅ (v : ↥K), (inner_right ↑v).ker

Kᗮ can be characterized as the intersection of the kernels of the operations of inner product with each of the elements of K.

source

theorem submodule.is_closed_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) :

is_closed ↑Kᗮ

The orthogonal complement of any submodule K is closed.

source

@[protected, instance]

def submodule.orthogonal.complete_space {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space E] :

complete_space ↥Kᗮ

In a complete space, the orthogonal complement of any submodule K is complete.

source

theorem submodule.orthogonal_gc (𝕜 : Type u_1) (E : Type u_2) [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

galois_connection submodule.orthogonal submodule.orthogonal

submodule.orthogonal gives a galois_connection between submodule 𝕜 E and its order_dual.

source

theorem submodule.orthogonal_le {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K₁ K₂ : submodule 𝕜 E} (h : K₁ ≤ K₂) :

K₂ᗮ ≤ K₁ᗮ

submodule.orthogonal reverses the ≤ ordering of two subspaces.

source

theorem submodule.orthogonal_orthogonal_monotone {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K₁ K₂ : submodule 𝕜 E} (h : K₁ ≤ K₂) :

K₁ᗮ ᗮ ≤ K₂ᗮ ᗮ

submodule.orthogonal.orthogonal preserves the ≤ ordering of two subspaces.

source

theorem submodule.le_orthogonal_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) :

K ≤ Kᗮ ᗮ

K is contained in Kᗮᗮ.

source

theorem submodule.inf_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K₁ K₂ : submodule 𝕜 E) :

K₁ᗮ ⊓ K₂ᗮ = (K₁ ⊔ K₂)ᗮ

The inf of two orthogonal subspaces equals the subspace orthogonal to the sup.

source

theorem submodule.infi_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {ι : Type u_3} (K : ι → submodule 𝕜 E) :

(⨅ (i : ι), (K i)ᗮ) = (supr K)ᗮ

The inf of an indexed family of orthogonal subspaces equals the subspace orthogonal to the sup.

source

theorem submodule.Inf_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (s : set (submodule 𝕜 E)) :

(⨅ (K : submodule 𝕜 E) (H : K ∈ s), Kᗮ) = (Sup s)ᗮ

The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup.

source

theorem submodule.sup_orthogonal_inf_of_is_complete {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K₁ K₂ : submodule 𝕜 E} (h : K₁ ≤ K₂) (hc : is_complete ↑K₁) :

K₁ ⊔ K₁ᗮ ⊓ K₂ = K₂

If K₁ is complete and contained in K₂, K₁ and K₁ᗮ ⊓ K₂ span K₂.

source

theorem submodule.sup_orthogonal_of_is_complete {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} (h : is_complete ↑K) :

K ⊔ Kᗮ = ⊤

If K is complete, K and Kᗮ span the whole space.

source

theorem submodule.sup_orthogonal_of_complete_space {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] :

K ⊔ Kᗮ = ⊤

If K is complete, K and Kᗮ span the whole space. Version using complete_space.

source

theorem submodule.exists_sum_mem_mem_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space ↥K] (v : E) :

∃ (y : E) (H : y ∈ K) (z : E) (H : z ∈ Kᗮ), v = y + z

If K is complete, any v in E can be expressed as a sum of elements of K and Kᗮ.

source

@[simp]

theorem submodule.orthogonal_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space ↥K] :

Kᗮ ᗮ = K

If K is complete, then the orthogonal complement of its orthogonal complement is itself.

source

theorem submodule.orthogonal_orthogonal_eq_closure {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space E] :

Kᗮ ᗮ = K.topological_closure

source

theorem submodule.is_compl_orthogonal_of_is_complete {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} (h : is_complete ↑K) :

is_compl K Kᗮ

If K is complete, K and Kᗮ are complements of each other.

source

@[simp]

theorem submodule.top_orthogonal_eq_bot {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

⊤ᗮ = ⊥

source

@[simp]

theorem submodule.bot_orthogonal_eq_top {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] :

⊥ᗮ = ⊤

source

@[simp]

theorem submodule.orthogonal_eq_bot_iff {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} (hK : is_complete ↑K) :

Kᗮ = ⊥ ↔ K = ⊤

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@[simp]

theorem submodule.orthogonal_eq_top_iff {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} :

Kᗮ = ⊤ ↔ K = ⊥

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theorem eq_orthogonal_projection_of_mem_orthogonal {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] {u v : E} (hv : v ∈ K) (hvo : u - v ∈ Kᗮ) :

↑(⇑(orthogonal_projection K) u) = v

A point in K with the orthogonality property (here characterized in terms of Kᗮ) must be the orthogonal projection.

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theorem eq_orthogonal_projection_of_mem_orthogonal' {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] {u v z : E} (hv : v ∈ K) (hz : z ∈ Kᗮ) (hu : u = v + z) :

↑(⇑(orthogonal_projection K) u) = v

A point in K with the orthogonality property (here characterized in terms of Kᗮ) must be the orthogonal projection.

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theorem orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space ↥K] {v : E} (hv : v ∈ Kᗮ) :

⇑(orthogonal_projection K) v = 0

The orthogonal projection onto K of an element of Kᗮ is zero.

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theorem orthogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] {K : submodule 𝕜 E} [complete_space E] {v : E} (hv : v ∈ K) :

⇑(orthogonal_projection Kᗮ) v = 0

The orthogonal projection onto Kᗮ of an element of K is zero.

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theorem orthogonal_projection_orthogonal_complement_singleton_eq_zero {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] [complete_space E] (v : E) :

⇑(orthogonal_projection (submodule.span 𝕜 {v})ᗮ) v = 0

The orthogonal projection onto (𝕜 ∙ v)ᗮ of v is zero.

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theorem eq_sum_orthogonal_projection_self_orthogonal_complement {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space E] [complete_space ↥K] (w : E) :

w = ↑(⇑(orthogonal_projection K) w) + ↑(⇑(orthogonal_projection Kᗮ) w)

In a complete space E, a vector splits as the sum of its orthogonal projections onto a complete submodule K and onto the orthogonal complement of K.

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theorem id_eq_sum_orthogonal_projection_self_orthogonal_complement {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (K : submodule 𝕜 E) [complete_space E] [complete_space ↥K] :

continuous_linear_map.id 𝕜 E = K.subtypeL.comp (orthogonal_projection K) + K

mathlib documentation

Inner Product Space #

Main results #

Notation #

Implementation notes #

TODO #

Tags #

References #

Constructing a normed space structure from an inner product #

Properties of inner product spaces #

Inner product space structure on product spaces #

Inner product space structure on subspaces #

Derivative of the inner product #

Continuity and measurability of the inner product #

Orthogonal projection in inner product spaces #