Consider two finite dimensional vector spaces $V_1,V_2$ and their duals denoted by $V_1^{*},V_{2}^{*}$. I am working on a problem that is asking me to prove a generalized version of the below, but I am completely lost.
There is a unique isomorphism $\Phi:V_1^{*} \otimes V_2^{*} \rightarrow (V_1\otimes V_2)^{*} $ such that $\Phi(f_1 \otimes f_2)(v_1 \otimes v_2) = f_1(v_1)f_2(v_2)$.
My goal is to eventually prove this for some finite $k$ vector spaces, but I cannot prove even this to begin with.
This is my first time working with tensor products and dual spaces. I would like some an explanation of what this means, and how to develop an intuition for this type of problem. Furthermore, and hints on how to get started with this question (without necessarily providing the entire answer) are appreciated.
I have previously worked on showing that we can define an antilinear isomorphism between $V$ and its dual as well as that the evaluation map provides an isomorphism for $V \rightarrow V^{**}$.
One thing that may help is to consider an explicit example of tensor products with dual spaces to get an idea of how they function. Consider the following:
Let $V$ be a vector space with finite basis $\{e_1,...,e_n\}$ and $V^*$ with the basis $\{\epsilon_1,...,\epsilon_n\}$ dual to $\{e_i\}$. Consider the isomorphism $$\alpha: V^*\otimes V \rightarrow Hom(V;V) $$ where for $\kappa \in V^*, \xi\in V$ one has $$V^*\otimes V \ni \kappa \otimes \xi \mapsto \kappa(\cdot)\xi \in Hom(V;V)$$ For a minute, take on faith that this is an isomorphism. Let $n=4$, $t = \epsilon_1 \otimes e_3 + \epsilon_4 \otimes e_2 \in V^*\otimes V$. Can you find $\alpha(t) \in Hom(V;V)$?
Generally I find that the easiest way to start working with tensor products is to make sure you really understand what elements of each set look like. Going back to $\Phi$ as defined in your question, let $\{e_i\}_{1\leq i \leq n}$ be a basis of $V_1$ with dual $\{\epsilon_i\}_{1\leq i \leq n}$ and similarly $\{b_i\}_{1\leq i \leq m}$ a basis of $V_2$ with dual $\{ \delta_i\}_{1\leq i \leq m}$. With this, suppose $$f = f_1\epsilon_1 + ... + f_n\epsilon_n \in V_1^* $$ $$g = g_1\delta_1 + ... + g_m\delta_m \in V_2^*$$ $$v_1 = a_1e_1+...+a_ne_n \in V_1$$ $$v_2 = c_1b_1+...+c_mb_m \in V_2$$ where each of $f_i, g_i, a_i,$ and $c_i$ are arbitrary elements of the underlying field. Can you find $\Phi(f\otimes g)$? What about $\Phi(f\otimes g)(v_1\otimes v_2)$? If you're having trouble, think about what set these items belong to, in order to glean some information on what they should look like.
While there is some fancier machinery I'm sure you could use, I'd recommend going back to the definition of a vector space isomorphism, and attempting to show that those properties hold for $\Phi$ for some fixed $n$ and $m$ in order to say that $\Phi$ is indeed an isomorphism.