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Understanding isomorphic equivalences of tensor product
I have the following question: Let $V$ be a vectorspace with an inner product $<.,.>$. Let $V^{*}$ be its dual. Is it true that $V \otimes V^{*} = End(V)$ ? If yes in which way ? what is the isomorpism ? Thanks in advance.
mika
You don't need the inner product. It's true more generally that if $V$ is a finite-dimensional vector space and $W$ is any vector space whatsoever, then $V^{\ast} \otimes W$ is canonically isomorphic to $\text{Hom}(V, W)$; that is, the two are naturally isomorphic functors $\text{FinVect}^{op} \times \text{Vect} \to \text{Vect}$ (so in particular they are isomorphic as $\text{GL}(V) \times \text{GL}(W)$-representations, which is a much stronger statement than that they are isomorphic as vector spaces). The isomorphism sends a pure tensor $f \otimes w$ to the map $$v \mapsto f(v) w$$
where $v \in V$.
