Theorems · Definition · functional analysis

TensorProduct.mapIsometry

{𝕜 : Type u_1} →
  {E : Type u_2} →
    {F : Type u_3} →
      {G : Type u_4} →
        {H : Type u_5} →
          [inst : RCLike 𝕜] →
            [inst_1 : NormedAddCommGroup E] →
              [inst_2 : InnerProductSpace 𝕜 E] →
                [inst_3 : NormedAddCommGroup F] →
                  [inst_4 : InnerProductSpace 𝕜 F] →
                    [inst_5 : NormedAddCommGroup G] →
                      [inst_6 : InnerProductSpace 𝕜 G] →
                        [inst_7 : NormedAddCommGroup H] →
                          [inst_8 : InnerProductSpace 𝕜 H] →
                            (E →ₗᵢ[𝕜] G) → (F →ₗᵢ[𝕜] H) → TensorProduct 𝕜 E F →ₗᵢ[𝕜] TensorProduct 𝕜 G H

The tensor product map of two linear isometries is a linear isometry. In particular, this is the linear isometry version of TensorProduct.map f g when f and g are linear isometries.

Defined in

Mathlib.Analysis.InnerProductSpace.TensorProduct

Cited by

9 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpace