Theorems · Definition · functional analysis

HilbertBasis.mk

{ι : Type u_1} →
  {𝕜 : Type u_2} →
    [inst : RCLike 𝕜] →
      {E : Type u_3} →
        [inst_1 : NormedAddCommGroup E] →
          [inst_2 : InnerProductSpace 𝕜 E] →
            [CompleteSpace E] →
              {v : ι → E} →
                Orthonormal 𝕜 v → ⊤ ≤ (Submodule.span 𝕜 (Set.range v)).topologicalClosure → HilbertBasis ι 𝕜 E

An orthonormal family of vectors whose span is dense in the whole module is a Hilbert basis.

Defined in

Mathlib.Analysis.InnerProductSpace.l2Space

Cited by

1 results in Mathlib

Foundations

Depth 232 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace