# vector spaces – Isomorphism between \$V^* otimes V\$ and End\$V\$

If $$V$$ is a vector space and End$$V$$ is the set of endomorphisms from $$V$$ to $$V$$.

Defining a map from $$V^*otimes Vrightarrow$$ End $$V$$ by sending some element say, $$fotimes v in V^* otimes V$$ to endomorphism whose value at $$win V$$ is $$f(w)v$$.

I am trying to show this map is an isomorphism between $$V^* otimes V$$ and End $$V$$. I am trying to verify it using dual basis.

If some hint can be provided, it will be a great help!