Theorems · Theorem · linear algebra

Matrix.posSemidef_opNorm_smul_gram_sub_gram

∀ {E : Type u_1} {n : Type u_2} {𝕜 : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] {F : Type u_5} [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace 𝕜 F]
  (v : n → E) (f : E →L[𝕜] F), (‖f‖ ^ 2 • Matrix.gram 𝕜 v - Matrix.gram 𝕜 (⇑f ∘ v)).PosSemidef

Inequality ‖f x‖ ≤ ‖f‖ * ‖x‖ lifted to Gram matrices.

Defined in

Mathlib.Analysis.InnerProductSpace.GramMatrix

Cited by

0 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpace