Theorems · Theorem · functional analysis

Submodule.orthogonal_orthogonal_eq_closure

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  (K : Submodule 𝕜 E) [CompleteSpace E], Kᗮᗮ = K.topologicalClosure

In a Hilbert space, the orthogonal complement of the orthogonal complement of a subspace K is the topological closure of K. Note that the completeness assumption is necessary. Let E be the space ℕ →₀ ℝ with inner space structure inherited from PiLp 2 (fun _ : ℕ ↦ ℝ). Let K be the subspace of sequences with the sum of all elements equal to zero. Then Kᗮ = ⊥, Kᗮᗮ = ⊤.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Cited by

3 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace