Isomorphism between Cartesian product of a vector space and the space of linear maps

Question

Let $V$ be a vector space defined over the field $\mathbb{F}$. Given a positive integer $n$, prove that $V^n = V \times V \cdots \times V$ ($n$ times) is isomorphic to $\mathcal{L}(\mathbb{F}^n, V)$ where $\mathcal{L}(\mathbb{F}^n, V)$ is the vector space of linear maps from $\mathbb{F}^n$ to $V$.

I am not sure how to approach this, especially since $V$ is not even assumed to be finite dimensional. What is the isomorphism between the two vector spaces?

Answer 1

An element $f\in\mathcal{L}(\mathbb{F}^n,V)$ is totally determined by its values on the canonical basis $(e_i)_{i=1,\dots,n}$ of $\mathbb{F}^n$. $$v_i = f(e_i)\in V.$$ You can check that $$f\longmapsto(f(e_1),\cdots,f(e_n))$$ is an isomorphism.