# mathlibdocumentation

analysis.analytic.linear

# Linear functions are analytic #

In this file we prove that a continuous_linear_map defines an analytic function with the formal power series f x = f a + f (x - a).

@[simp]
noncomputable def continuous_linear_map.fpower_series {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (x : E) :
F

Formal power series of a continuous linear map f : E βL[π] F at x : E: f y = f x + f (y - x).

Equations
@[simp]
theorem continuous_linear_map.fpower_series_apply_add_two {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (x : E) (n : β) :
(n + 2) = 0
@[simp]
theorem continuous_linear_map.fpower_series_radius {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (x : E) :
@[protected]
theorem continuous_linear_map.has_fpower_series_on_ball {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (x : E) :
@[protected]
theorem continuous_linear_map.has_fpower_series_at {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (x : E) :
x
@[protected]
theorem continuous_linear_map.analytic_at {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] (f : E βL[π] F) (x : E) :
analytic_at π βf x
noncomputable def continuous_linear_map.uncurry_bilinear {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {G : Type u_4} [normed_space π G] (f : E βL[π] F βL[π] G) :
(Ξ» (i : fin 2), E Γ F) G

Reinterpret a bilinear map f : E βL[π] F βL[π] G as a multilinear map (E Γ F) [Γ2]βL[π] G. This multilinear map is the second term in the formal multilinear series expansion of uncurry f. It is given by f.uncurry_bilinear ![(x, y), (x', y')] = f x y'.

Equations
@[simp]
theorem continuous_linear_map.uncurry_bilinear_apply {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {G : Type u_4} [normed_space π G] (f : E βL[π] F βL[π] G) (m : fin 2 β E Γ F) :
@[simp]
noncomputable def continuous_linear_map.fpower_series_bilinear {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {G : Type u_4} [normed_space π G] (f : E βL[π] F βL[π] G) (x : E Γ F) :
(E Γ F) G

Formal multilinear series expansion of a bilinear function f : E βL[π] F βL[π] G.

Equations
@[simp]
theorem continuous_linear_map.fpower_series_bilinear_radius {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {G : Type u_4} [normed_space π G] (f : E βL[π] F βL[π] G) (x : E Γ F) :
@[protected]
theorem continuous_linear_map.has_fpower_series_on_ball_bilinear {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {G : Type u_4} [normed_space π G] (f : E βL[π] F βL[π] G) (x : E Γ F) :
@[protected]
theorem continuous_linear_map.has_fpower_series_at_bilinear {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {G : Type u_4} [normed_space π G] (f : E βL[π] F βL[π] G) (x : E Γ F) :
has_fpower_series_at (Ξ» (x : E Γ F), β(βf x.fst) x.snd) x
@[protected]
theorem continuous_linear_map.analytic_at_bilinear {π : Type u_1} {E : Type u_2} [normed_space π E] {F : Type u_3} [normed_space π F] {G : Type u_4} [normed_space π G] (f : E βL[π] F βL[π] G) (x : E Γ F) :
analytic_at π (Ξ» (x : E Γ F), β(βf x.fst) x.snd) x