Let $V,W$ be vector spaces. Prove that $(V\times W)^{*}$ is isomorphic to $V^{*}\times W^{*}.$ I know that this question has been asked before. However, my objective is to know whether my proof is correct or not since I am new to this subject.
Let $V,W$ be vector spaces. Prove that $(V\times W)^{*}$ is  isomorphic to 
$V^{*}\times W^{*}.$
Let $\psi:(V\times W)^{*}\to V^*\times W^*.$ Then $\psi(\phi(v,w))=(\phi(v,0),\phi(0,w))$ for any $v\in V$ and $w\in W.$ 
Now we have to show that this map is surjective and injective. Let $\psi(\phi)=\psi(\chi)$ for any $\phi,\chi\in (V\times W)^*.$ Then we want to show that $\phi=\chi.$ Now for any $v\in V$ and $w\in W$ we have that $\psi(\phi(v,w))=(\phi(v,0),\phi(0,w))=(\chi(v,0),\chi(0,w)).$ Thus $\phi(v,0)=\chi(v,0)$ and $\phi(0,w)=\chi(0,w).$ Adding both these equalities we get $\phi(v,w)=\chi(v,w)$ and so $\chi=\phi.$
Now we want to show surjectivity. Let $(f,g)\in V^*\times W^*.$ We define $\psi (v,0)=f(v)$ for all $v\in V$ and $\psi(0,w)=g(w)$ for all $w\in W.$ Then we have that $\psi(v,w)=f(v)+g(w).$ And so for any tuple $(f,g)$ we have a linear functional in $(V\times W)^*.$ 
We also have to show that $\psi$ preserves addition and scalar multiplication. Thus consider $\phi,\chi\in (V\times W)^*$ and $\gamma\in \mathbb{K}$, then we have 
$\psi(\phi+\gamma \chi)(v,w)=((\phi+\gamma \chi) (v,0),(\phi+\gamma \chi)(0,w))$
$=(\phi(v,0)+\chi(\gamma v,0),\phi(0,w)+\chi(0,\gamma w))=(\phi(v,0),\phi(0,w))+\gamma(\chi(v,0),\chi(0,w)).$ 
Which shows that the map is linear and hence it is an isomorphism.
 A: The idea is right, but the map $\psi$ is not well defined.
You want $\Psi\colon(V\times W)^*\to V^*\times W^*$. Let $\phi\in(V\times W)^*$; define $\Psi(\phi)$ to be $(\phi_V,\phi_W)$, where
$$
\phi_V(v)=\phi(v,0)
\qquad
\phi_W(w)=\phi(0,w)
$$
The value of $\Psi$ at $\phi$ must be a pair $(\xi,\eta)\in V^*\times W^*$, not an element of $V\times W$.
Now prove that $\Psi$ is linear: if $\alpha,\beta\in(V\times W)^*$ and $a,b\in\mathbb{K}$, you want $\Psi(a\alpha+b\beta)=a\Psi(\alpha)+b\Psi(\beta)$, which is the same as proving that
$$
(a\alpha+b\beta)_V=a\alpha_V+b\beta_V
\qquad\text{and}\qquad
(a\alpha+b\beta)_W=a\alpha_W+b\beta_W
$$
(prove it).
Next you need to prove that $\ker\Psi=\{0\}$ (it should be easy) and that $\Psi$ is surjective. If $\xi\in V^*$ and $\eta\in W^*$, define
$$
\phi\colon V\times W\to\mathbb{K}
\qquad
\phi(v,w)=\xi(v)+\eta(w)
$$
Prove that $\phi\in(V\times W)^*$ and $\Psi(\phi)=(\xi,\eta)$.
A: To complement @egreg's answer, on can take the categorical point of view. 
Consider the following diagram:

where $p_V, p_W$ are the canonical projections from $V\times W$ onto its factors,  $i_V, i_W$ the canonical injections and $K$ the base field.
The inverse homomorphisms are defined this way:
\begin{align}
(V\times W)^*&\longrightarrow V^*\times W^*\\
\phi& \longmapsto (\phi\circ i_V,\phi\circ i_W) \\
\phi_V\circ p_V+\phi_W\circ p_W&\longleftarrow(\phi_V,\phi_W)
\end{align}
and it's easy to check that composing these maps both ways yields the identity map in each case.
