Theorems · Theorem · functional analysis

IsHilbertSum.linearIsometryEquiv_symm_apply_dfinsupp_sum_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] [inst_7 : (i : ι) → DecidableEq (G i)] (hV : IsHilbertSum 𝕜 G V) (W₀ : Π₀ (i : ι), G i),
  hV.linearIsometryEquiv.symm (W₀.sum (lp.single 2)) = W₀.sum fun i => ⇑(V i)

In the canonical isometric isomorphism between a Hilbert sum E of G : ι → Type* and lp G 2, a finitely-supported vector in lp G 2 is the image of the associated finite sum of elements of E.

Defined in

Mathlib.Analysis.InnerProductSpace.l2Space

Cited by

1 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceCompleteSpaceDecidableEqDecidableEq