I encountered the following fact: let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, and consider the tensor product $\mathfrak{g}\otimes_{\mathbb{C}} C^{\infty}(S^1)$ (where I guess we mean maps from $S^1$ to $\mathbb{C}$ and not to $\mathbb{R}$, as one would be inclined to think). Then the following holds: $$\mathfrak{g}\otimes C^{\infty}(S^1) \cong C^{\infty}(S^1;\mathfrak{g})$$ I tried using the obvious map which sends $x\otimes f$ to the map $p\mapsto f(p)x$, but I can't see how this is a bijection.
I think this holds in general for a vector space $V$ instead of $\mathfrak{g}$, but I don't think this helps. I'd be grateful if someone can help me.