Reference for a proof: $\mathsf{Hom}(V, W)\longrightarrow \mathsf{Hom}(W^*, V^*), f\longmapsto f^*$, is an isomorsphim. Let $V$ and $W$ be two vector spaces over a field $\mathbb K$. The transpose of a linear map $f:V\longrightarrow W$ is the linear map $f^*:W^*\longrightarrow V^*$ given by $$f^*(u):=u\circ f.$$ I read in this page that whenever $V$ or $W$ are finite dimensional then the map $$\mathsf{Hom}(V, W)\longrightarrow \mathsf{Hom}(W^*, V^*), f\longmapsto f^*,$$ is an isomorphism. I looked for the proof in some standard books and couldn't find it, can anyone recomend me a reference where I can find the proof?
Thanks.
 A: If $V$ is finite-dimensional, then $V\cong \mathbb{K}^m$ for some nonnegative integer $m$.  Then, $$\text{Hom}_\mathbb{K}(V,W)\cong \text{Hom}_\mathbb{K}\left(\mathbb{K}^m,W\right)\cong \Big(\text{Hom}_\mathbb{K}\left(\mathbb{K},W\right)\Big)^m \cong W^m\,.$$
On the other hand, $$\text{Hom}_\mathbb{K}\left(W^*,V^*\right)\cong\text{Hom}_\mathbb{K}\left(W^*,\mathbb{K}^m\right)\cong\Big(\text{Hom}_\mathbb{K}\left(W^*,\mathbb{K}\right)\Big)^m\cong \left(W^{**}\right)^m\,.$$
Unless $W$ is finite-dimensional, $W^m$ is not isomorphic to $\left(W^{**}\right)^m$.
If $W$ is finite-dimensional, then $W\cong \mathbb{K}^n$ for some nonnegative integer $n$.  Then,
$$\text{Hom}_\mathbb{K}(V,W)\cong \text{Hom}_\mathbb{K}\left(V,\mathbb{K}^n\right)\cong \Big(\text{Hom}_\mathbb{K}\left(V,\mathbb{K}\right)\Big)^n\cong\left(V^*\right)^n\,.$$
On the other hand,
$$\text{Hom}_\mathbb{K}\left(W^*,V^*\right)\cong\text{Hom}\left(\mathbb{K}^n,V^*\right)\cong\Big(\text{Hom}_\mathbb{K}\left(\mathbb{K},V^*\right)\Big)^n\cong \left(V^{*}\right)^n\,.$$
Thus, as Bye-World mentioned, $\text{Hom}_\mathbb{K}(V,W)\cong \text{Hom}_\mathbb{K}\left(W^*,V^*\right)$ if and only if $W$ is finite-dimensional (regardless of the dimension of $V$).

What happens when $V$ and $W$ are both infinite-dimensional?
Let $m:=\dim_\mathbb{K}V$ and $n:=\dim_\mathbb{K}W$, where $m$ and $n$ are cardinal numbers (not necessarily finite).  Then,
$$\text{Hom}_\mathbb{K}(V,W)\cong \text{Hom}_\mathbb{K}\left(\mathbb{K}^{\oplus m},W\right)\cong \Big(\text{Hom}_\mathbb{K}\left(\mathbb{K},W\right)\Big)^{\times m} \cong W^{\times m}\,.$$
Here, for a vector space $X$, the notation $X^{\times k}$ denotes the direct product $\prod_{i\in J}\,X$, where $J$ is an index set with $|K|=k$.  Similarly, $X^{\oplus k}$ denotes the direct sum $\bigoplus_{i\in J}\,X$.  
Note that $\dim_\mathbb{K}\left(X^{\times k}\right)=|X|^k$ for an infinite-dimensional vector space $X$ and for an infinite cardinal $k$ (see here).  In particular, $\dim_\mathbb{K}X^*=|X|=|\mathbb{K}|^{\dim_\mathbb{K}X}$.  Thus,
$$\dim_\mathbb{K}\text{Hom}_\mathbb{K}(V,W)=|W|^{m}=\big(|\mathbb{K}|\,n\big)^m=\big(\max\{\kappa,n\}\big)^m=\max\left\{\kappa^m,n^m\right\}\,,$$
where $\kappa:=|\mathbb{K}|$.
On the other hand,
$$\text{Hom}_\mathbb{K}\left(W^*,V^*\right)\cong\text{Hom}\left(\mathbb{K}^{\oplus (\kappa^n)},V^*\right)\cong\Big(\text{Hom}_\mathbb{K}\left(\mathbb{K},V^*\right)\Big)^{\times (\kappa^n)}\cong \left(V^{*}\right)^{\times (\kappa^n)}\,.$$
Consequently, 
$$\dim_\mathbb{K}\text{Hom}_\mathbb{K}\left(W^*,V^*\right)=\left|V^*\right|^{\kappa^n}=\Big(\kappa^{m}\Big)^{\kappa^n}=\kappa^{m(\kappa^n)}=\kappa^{\max\{m,\kappa^n\}}=\max\left\{\kappa^m,\kappa^{\kappa^n}\right\}\,.$$
A: The map is injective. If $f\ne0$, take $v\in V$ with $w=f(v)\ne0$. Then there exists $u\in W^*$ with $u(w)\ne0$ (complete $w$ to a basis of $W$). Therefore $f^*(u)\ne0$, because $u\circ f(v)=u(w)\ne0$.
Suppose $V$ and $W$ are finite dimensional. Then the existence of an injective map $\operatorname{Hom}(V,W)\to\operatorname{Hom}(W^*,V^*)$ implies
$$
\dim\operatorname{Hom}(V,W)\le\dim\operatorname{Hom}(W^*,V^*)
$$
(it's easy to see $\operatorname{Hom}(V,W)$, $V^*$ and $W^*$ are finite dimensional as well).
By the same reason,
$$
\dim\operatorname{Hom}(W^*,V^*)\le\dim\operatorname{Hom}(V^{**},W^{**})
$$
Now we can easily build an isomorphism
$$
\operatorname{Hom}(V^{**},W^{**})\to\operatorname{Hom}(V,W)
$$
by using the fact that the canonical maps $V\to V^{**}$ and $W\to W^{**}$ are isomorphisms.

An alternative proof for surjectivity in the finite dimensional case.
Fix a basis of $W$, say $\{w_1,\dots,w_n\}$ and consider that a map $g\colon W^*\to V^*$ is determined by its action on the dual basis $\{w_1^*,\dots,w_n^*\}$. This means that, given arbitrary elements $u_1,\dots,u_n\in V^*$, we need to find $f\colon V\to W$ such that
$$
f^*(w_i^*)=u_i,\qquad i=1,\dots,n
$$
This means $w_i^*\circ f=u_i$. Let $\{v_1,\dots,v_m\}$ be a basis of $V$. We must have $w_i^*(f(v_j))=u_i(v_j)$.
Writing $f(v_j)=\sum_{k=1}^n \alpha_{kj}w_k$ this amounts to
$$
\alpha_{ij}=u_i(v_j)
$$
and we can so define $f$ with the required property.
