Theorems · Theorem · functional analysis

IsHilbertSum.linearIsometryEquiv_symm_apply_single

∀ {ι : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] {G : ι → Type u_4} [inst_3 : (i : ι) → NormedAddCommGroup (G i)]
  [inst_4 : (i : ι) → InnerProductSpace 𝕜 (G i)] [inst_5 : CompleteSpace E] {V : (i : ι) → G i →ₗᵢ[𝕜] E}
  [inst_6 : DecidableEq ι] (hV : IsHilbertSum 𝕜 G V) {i : ι} (x : G i),
  hV.linearIsometryEquiv.symm (lp.single 2 i x) = (V i) x

In the canonical isometric isomorphism between a Hilbert sum E of G : ι → Type* and lp G 2, an "elementary basis vector" in lp G 2 supported at i : ι is the image of the associated element in E.

Defined in

Mathlib.Analysis.InnerProductSpace.l2Space

Cited by

2 results in Mathlib

Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceCompleteSpaceDecidableEq