# mathlibdocumentation

measure_theory.l2_space

# L^2 space #

If E is an inner product space over 𝕜 (ℝ or ℂ), then Lp E 2 μ (defined in lp_space.lean) is also an inner product space, with inner product defined as inner f g = ∫ a, ⟪f a, g a⟫ ∂μ.

### Main results #

• mem_L1_inner : for f and g in Lp E 2 μ, the pointwise inner product λ x, ⟪f x, g x⟫ belongs to Lp 𝕜 1 μ.
• integrable_inner : for f and g in Lp E 2 μ, the pointwise inner product λ x, ⟪f x, g x⟫ is integrable.
• L2.inner_product_space : Lp E 2 μ is an inner product space.
theorem measure_theory.L2.snorm_rpow_two_norm_lt_top {α : Type u_1} {F : Type u_3} {μ : measure_theory.measure α} [normed_group F] [borel_space F] (f : μ)) :
measure_theory.snorm (λ (x : α), f x ^ 2) 1 μ <
theorem measure_theory.L2.snorm_inner_lt_top {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [is_R_or_C 𝕜] {μ : measure_theory.measure α} [ E] [borel_space E] (f g : μ)) :
measure_theory.snorm (λ (x : α), inner (f x) (g x)) 1 μ <
@[protected, instance]
def measure_theory.L2.measure_theory.Lp.has_inner {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [is_R_or_C 𝕜] {μ : measure_theory.measure α} [ E] [borel_space E] [borel_space 𝕜] :
μ)
Equations
theorem measure_theory.L2.inner_def {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [is_R_or_C 𝕜] {μ : measure_theory.measure α} [ E] [borel_space E] [borel_space 𝕜] (f g : μ)) :
g = (a : α), inner (f a) (g a) μ
theorem measure_theory.L2.integral_inner_eq_sq_snorm {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [is_R_or_C 𝕜] {μ : measure_theory.measure α} [ E] [borel_space E] [borel_space 𝕜] (f : μ)) :
(a : α), inner (f a) (f a) μ = ((∫⁻ (a : α), (nnnorm (f a)) ^ 2 μ).to_real)
theorem measure_theory.L2.mem_L1_inner {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [is_R_or_C 𝕜] {μ : measure_theory.measure α} [ E] [borel_space E] [borel_space 𝕜] (f g : μ)) :
measure_theory.ae_eq_fun.mk (λ (x : α), inner (f x) (g x)) _ μ
theorem measure_theory.L2.integrable_inner {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [is_R_or_C 𝕜] {μ : measure_theory.measure α} [ E] [borel_space E] [borel_space 𝕜] (f g : μ)) :
measure_theory.integrable (λ (x : α), inner (f x) (g x)) μ
@[protected, instance]
def measure_theory.L2.inner_product_space {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [is_R_or_C 𝕜] {μ : measure_theory.measure α} [ E] [borel_space E] [borel_space 𝕜] :
μ)
Equations