Theorems · Definition · functional analysis

OrthogonalFamily.linearIsometry

{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)] →
                  [CompleteSpace E] → {V : (i : ι) → G i →ₗᵢ[𝕜] E} → OrthogonalFamily 𝕜 G V → ↥(lp G 2) →ₗᵢ[𝕜] E

A mutually orthogonal family of subspaces of E induce a linear isometry from lp 2 of the subspaces into E.

Defined in

Mathlib.Analysis.InnerProductSpace.l2Space

Cited by

11 results in Mathlib

Foundations

Depth 226 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceCompleteSpace