Theorems · Theorem · functional analysis

LinearIsometryEquiv.inner_map_map

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {E' : Type u_7} [inst_3 : SeminormedAddCommGroup E'] [inst_4 : InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (x y : E),
  inner 𝕜 (f x) (f y) = inner 𝕜 x y

A linear isometric equivalence preserves the inner product.

Defined in

Mathlib.Analysis.InnerProductSpace.LinearMap

Cited by

14 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpaceSeminormedAddCommGroupInnerProductSpace