Let's say you have a vector space $V$ and a vector space $W$ over the same field $k$. Then, according to a huge pile of books on representation theory I read, there is an obvious isomorphism $V^\ast\otimes_k W \cong \mathrm{Hom}_k(V,W)$, where $V^\ast=\mathrm{Hom}_k(V,k)$. I concur, it's not hard to write down, but then why don't they just write it down? It is given by
\[
\begin{array}{rcl}
\phi: V^\ast\otimes_k W &\longrightarrow& \mathrm{Hom}_k(V,W) \\
f\otimes w &\longmapsto& (v\mapsto f(v)w),
\end{array}\]
when we assume $W$ to be finite-dimensional. To construct an inverse, simply choose a basis $B$ for $W$ and consider the map that sends $g\in\mathrm{Hom}_k(V,W)$ to
\[ \psi(g) := \sum_{b\in B} (b^\ast\circ g)\otimes b \]
Then, note that $\sum_{b\in B} b^\ast(w)b = w$ for all $w\in W$ - simply by the definition of the dual basis, so
\[
\begin{array}{rcl}
\phi(\psi(g))(v) &=& \sum_{b\in B} b^\ast(g(v))\cdot b = g(v) \\
\psi(\phi(f\otimes w))&=&\sum_{b\in B} (b^\ast\circ(v\mapsto f(v)w)\otimes b
= \sum_{b\in B} (b^\ast(w)\cdot f)\otimes b
\\ &=& f \otimes \sum_{b\in B} b^\ast(w)b = f\otimes w.
\end{array}\]
We can now extend this result to show that
\[
V_1^\ast \otimes \cdots \otimes V_n^\ast \otimes W \cong L_n(V_1,\ldots,V_n; W),
\]
the space of multilinear maps $V_1\times\cdots\times V_n \to W$. In fact, this is best done by induction: Simply consider the (obvious) isomorphism
\[
\begin{array}{rcl}
L_{n-1}(V_1,\ldots,V_{n-1}; \mathrm{Hom}_k(V_n, W)) &\longrightarrow& L_n(V_1,\ldots,V_n; W)\\
f &\longrightarrow& ( (v_1,\ldots,v_n)\mapsto f(v_1,\ldots,v_{n-1})(v_n)) \\
g(v_1,\ldots,v_{n-1},\,\cdot\,) &\longleftarrow& g
\end{array} \] and use it for the induction step.