TITLE: Quick question: tensor product and dual of vector space
QUESTION [11 upvotes]: Recall that for a finite dimensional vector space $V$ we have the natural isomorphism $\phi :V^{*} \otimes V \rightarrow Hom(V,V)$ given by $\alpha \otimes v \mapsto (x \mapsto \alpha (x)v)$.
Is there a coordinate-free way to express $\phi^{-1}(id_{V})$?

REPLY [2 votes]: The required object, $\phi^{-1}(id_V)$, doesn't exist when $V$ is infinite dimensional. So any way of expressing it has to apply only when $V$ is fin-dim. The definition of "finite dimensional" is "has a finite basis". So you are always going to need a basis somewhere.