Consider $V$ an infinite-dimensional vector space over a field $F$ (here $F$ may be finite or infinite). Then is the space $\mathcal{L}(F,V) \cong V ?$ I think that this might be false. I also have an example where above conditions are true and $\mathcal{L}(V,F) \ncong V$ but how to show it for $\mathcal{L}(F, V)$? any hints/ideas? Am I missing something imortant? Please help thanks.
PS: $\mathcal{L}(X,Y) :=$ the space of all linear transformations from $X$ to $Y$, where $X$ and $Y$ are vector spaces.
An $F$-linear map $\Phi$ from $F$ into an $F$-vector space $V$ is uniquely determined by its value $\Phi(1)$. For $\lambda \in F$ we have $\Phi(\lambda) = \lambda \Phi(1)$ by $F$-linearity. This induces an isomorphism between $V$ and $\mathcal{L}(F,V)$.