Is there a name for this "co-dual" vector space? Let $V$ be a $K$-vector space, then the linear transformations $T:K\rightarrow V$ (where $K$ is considered a 1-dimensional vector space over itself) form a vector space
$$ A = \mathrm{span}\{T:K\rightarrow V\} $$
which is isomorphic to $V$.
The dual space is of course $V^* = \mathrm{span}\{T:V\rightarrow K\}$, but is there a name for this "co-dual space" $A$? (I'm using the "co-" here rather loosely.)
 A: There is no reason to give a name to the $K$-space $\text{Hom}_K(K,V)$. It is canonically isomorphic to $V$. Now, while canonical has a technical meaning, here it has a very concrete one. The isomorphism $\text{Hom}_K(K,V)\xrightarrow{\approx}V$ has a trivial, natural, satisfying form:
$$f\mapsto f(1)$$
Think of the elements of $\text{Hom}_K(K,V)$ as just "picking out" an element of $V$. Indeed, an element $f\in\text{Hom}_K(K,V)$ is completely determined by where it sends $1$, and it can send $1$ wherever it wants. Thus, it almost makes sense to think of $\text{Hom}_K(K,V)$ as the following set
$$\text{Hom}_K(K,V)=\left\{\{\text{I pick }v\}:v\in V\right\}$$
When thought about this way, what is the $K$-space operation on $\text{Hom}_K(K,V)$? Well, since $(\alpha f+\beta g)(1)=\alpha f(1)+\beta g(1)$ it's merely
$$\alpha\{\text{I pick }v\}+\beta\{\text{I pick }w\}=\{\text{I pick }\alpha v+\beta w\}$$
If you ignore the superfluous "I pick"s then, this vector space is, as naturally as you'd like $V$.
To pound it in, just a little bit more, think about how silly it would be to name the set of all set maps $\{\text{pt}\}\to X$, for some set $X$. This is JUST $X$. 
