Theorems · Theorem · functional analysis

Submodule.norm_sq_eq_add_norm_sq_projection

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  (x : E) (S : Submodule 𝕜 E) [inst_3 : S.HasOrthogonalProjection],
  ‖x‖ ^ 2 = ‖S.orthogonalProjectionOnto x‖ ^ 2 + ‖Sᗮ.orthogonalProjectionOnto x‖ ^ 2

The Pythagorean theorem, for an orthogonal projection.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

1 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection