Prove $V_1^*\otimes V_2^*$ is isomorphic to $(V_1\otimes V_2)^*$ Good morning. I need prove this:
Prove $V_1^*\otimes V_2^*$ is isomorphic to $(V_1\otimes V_2)^*$
For prove this, i need prove injectivity and surjectivity of a function, But i'm stcuk trying to define the function.
Can someone help me?
 A: Assume $V_1$ and $V_2$ are finite-dimensional.
Let $\{e_1, \ldots, e_n\}$ be a basis for $V_1$, and $\{e_1, \ldots, f_m\}$ be a basis for $V_2$. Let $\{e_1^*, \ldots, e_n^*\}$ and $\{f_1^*, \ldots, f_m^*\}$ be the dual bases.
Then $\{e_i^* \otimes f_j^*\}_{i = 1, \ldots, n;\, j=1, \ldots, m}$ is a basis for $V_1^* \otimes V_2^*$.
On the other hand, $\{e_i \otimes f_j\}_{i = 1, \ldots, n;\, j=1, \ldots, m}$ is a basis for $V_1 \otimes V_2$ so its dual basis $\{(e_i \otimes f_j)^*\}_{i = 1, \ldots, n;\, j=1, \ldots, m}$ is a basis for $(V_1 \otimes V_2)^*$.
The linear map given by $e_i^* \otimes f_j^* \mapsto (e_i \otimes f_j)^*$ is an obvious isomorphism.
On the other hand, if $V_1 \otimes V_2$ is infinite-dimensional and has a basis $\{e_n\}_{n=1}^\infty$ then its dual "basis" $\{e_n^*\}_{n=1}^\infty$ is in fact not a basis, since for example $\sum_{n=1}^\infty e_n^*$ is also a linear map on $V_1 \otimes V_2$.
A: This is true only for finite-dimensional vector spaces.
Hint:
Consider the composition of the following homomorphisms ($K$ denotes the base field): 


*

*$\DeclareMathOperator{\Hom}{Hom}
\varphi:(V_1\otimes_K V_2)^*=\Hom_K(V_1\otimes_K V_2,K)\longrightarrow \Hom_K(V_1, V_2^*)$


defined  for  each $f\in(V_1\otimes_K V_2)^* \,$ by
$$\varphi(f)(v_1):v_2\in V_2\longmapsto f(v_1\otimes v_2).$$
Check this is an isomorphism (no dimension hypothesis required here).


*

*$V_1^*\otimes_K W\longrightarrow \Hom_K(V_1, W)$, defined by 
$$(\varphi_1,w)\longmapsto \bigl(v_1\longmapsto \varphi_1(v_1\otimes w)\bigr).$$
This one requires $W$ to be finite-dimensional. 


Let $(e_i)_{1\le i\le n}$  a basis of $V_1$, and $(e_i^*)$ the dual basis in $V_1^*$. You can check the inverse isomorphism is defined, for any  linear map $u$ from $V_1$ to $W$, by the following formula:
$$u\longmapsto \sum_{i=1}^n e_i^*\otimes u(e_i).$$
