Let $V,W$ be vector space over field $k$. Then \begin{eqnarray*} V^*\otimes W &=&V^*(\oplus_{i\in I} k_i)\\ &=&\oplus_{i\in I}(V^*\otimes k_i)\\ &=& \oplus_{i\in I}V^*_i\\ &=& \oplus_{i\in I}Hom(V,k)_i, \end{eqnarray*} \begin{eqnarray*} Hom(V,W)=Hom(V,\oplus_{i\in I}k_i). \end{eqnarray*} When $W$ is finite dimensional, we have $Hom(V,\oplus_{i\in I}k_i)=\oplus_{i\in I}Hom(V,k)_i$. Thus $V^*\otimes W=Hom(V,W)$.
Does the equality holds when $W$ is infinite dimensional and $V$ finite dimensional? Is the equality holds when $W$ is infinite dimensional and $V$ also infinite dimensional?
I am confused. By universal property I only obtain $ Hom(M,\Pi_{i\in I}N_i)=\Pi_{i\in I}Hom(M,N_i)$. Can we obtain $Hom(M,\oplus_{i\in I}N_i)=\oplus_{i\in I}Hom(M,N_i)$ in general?
Thanks.
