# Cotangent bundle tensor product tangent bundle

What is the meaning of Cotangent bundle tensor product tangent bundle:

$T^*M\otimes TM$? what will an element of this space be?

This is isomorphic to $Hom(TM,TM)$. Namely it is the endomorphism bundle of the tangent bundle. In general given two vector spaces(or bundles) we have an isomorphism between $W \otimes V^*$ and $Hom(V,W)$.
• You need $V$ to be finite-dimensional here. – Qiaochu Yuan Jan 20 '16 at 7:28
• @QiaochuYuan Or for $W$ to be finite, no? Either way every element of $\mathrm{Hom}(V,W)$ has finite rank. – Oscar Cunningham Jan 21 '16 at 11:29