What is the dual space for the Hom vector space $\text{Hom}(V,W)$?

Question

Consider, a hom-set $\text{Hom}(V,W) = \{\phi | \phi: V \to W\}$, where $V$ and $W$ both are vector spaces. This $\text{Hom}(V,W)$ can be made into a vector space over the field $K$ by defining scalar multiplication and vector addition map. Now I am trying to know what is the dual space of this $\text{Hom}(V,W)$. According to the definition of the dual space, we construct it as a set of linear maps, $f's$, that accept an element of $\text{Hom}(V,W)$ and output an element from the field $K$, $f: \text{Hom}(V,W) \to K$. Since $\text{Hom}(V,W)$ is itself a linear map from $V$ to $W$, can I define an element ($f$) of the dual space as $f: V \to W \to K$, implying the $f$ as $f: V \to K$, which is exact the dual space of $V$. Hence, this leads to an isomorphism between $\text{Hom}(V,W)^*$ and $V^*$. Actually, I am not sure about this formulation and want to know is my approach correct to define $\text{Hom}(V,W)^*$ as isomorphic to $V^*$?

Answer 1

You can identify $Hom(V,W)$ as $V^* \otimes W$ via the isomorphism $$ \phi \otimes w \mapsto [v \mapsto \phi(v)w] $$ where $\phi \in V^*, w \in W$ and $v \in V$. Moreover, it's not difficult to get convinced that $(V \otimes W)^* \simeq V^* \otimes W^*$ in a canonical way.

Therefore, $$ Hom(V,W)^* \simeq (V^* \otimes W)^* \simeq V^{**} \otimes W^*. $$ For generic vector spaces you cannot say much more, however, if $V$ is reflective (e.g. if it is finite dimensional), then $V^{**} \simeq V$ canonically and so you may find $$ Hom(V,W)^* \simeq V \otimes W^* \simeq Hom(W,V) $$