Theorems · Theorem · functional analysis

TensorProduct.ext_iff_inner_right

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace 𝕜 F]
  {x y : TensorProduct 𝕜 E F}, x = y ↔ ∀ (a : E) (b : F), inner 𝕜 x (a ⊗ₜ[𝕜] b) = inner 𝕜 y (a ⊗ₜ[𝕜] b)

Given x, y : E ⊗ F, x = y iff ⟪x, a ⊗ₜ b⟫ = ⟪y, a ⊗ₜ b⟫ for all a, b.

Defined in

Mathlib.Analysis.InnerProductSpace.TensorProduct

Cited by

1 results in Mathlib

Foundations

Depth 244 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpace