Lagmental Vicfred

The Tensor--Hom Adjunction Curries Bilinear Data by Vicfred

tags: mathematics, multilinear algebra, tensor products, matrix functions

Linear maps from a tensor product correspond naturally to linear maps into a Hom space. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Objects and notation

Tensor and exterior powers turn multilinear behavior into linear maps. For finite-dimensional \(V\), the spaces \(V^{\otimes k}\) and \(\bigwedge^kV\) carry induced actions of every \(T\in\operatorname{End}(V)\).

$$ U,V,W\ \text{finite-dimensional} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \operatorname{Hom}(U\otimes V,W)\cong\operatorname{Hom}(U,\operatorname{Hom}(V,W)) $$

Push the symbols

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \Phi(u\otimes v)=B(u,v)\quad\longleftrightarrow\quad\widetilde\Phi(u)(v)=B(u,v) $$

Structural reading

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad U,V,W\ \text{finite-dimensional},\\[5pt] \mathsf{C}\;&:\quad \operatorname{Hom}(U\otimes V,W)\cong\operatorname{Hom}(U,\operatorname{Hom}(V,W)). \end{aligned} $$

A hypothesis worth keeping

Tensor coordinates depend on a basis even when the tensor does not. Index notation is safe only when contraction rules and variance are clear.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Hom}(U\otimes V,W)\cong\operatorname{Hom}(U,\operatorname{Hom}(V,W)) \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sat 04 July 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.