Let $V$ be a vector space generated by $v_1,\ldots,v_n$ over a field $\mathbb{K}$ ($\mathbb{R}$ or $\mathbb{C}$) and let $V^*$ be its dual generated by the corresponding $v^1,\ldots,v^n$. Let $W$ be a vector space generated by $w_1,\ldots,w_m$ over $\mathbb{K}$ and let $W^*$ be its dual generated by the corresponding $w^1,\ldots,w^m$.
Consider the map $f:V^*\bigotimes_\mathbb{K} W\longrightarrow\text{Hom}_\mathbb{K}(V,W)$ defined by $f(\beta)x=\sum_{k=1}^m\beta(\iota_x,w^k)w_k$ for all $x\in V$, where $\iota_x:V^*\longrightarrow\mathbb{K}$ is the linear map defined by $\iota_x(f)=f(x)$ for all $f\in V^*$.
Consider also the map $g:\text{Hom}_\mathbb{K}(V,W)\longrightarrow V^*\bigotimes_\mathbb{K} W$ defined by $g(\alpha)=\sum_{j=1}^n v^j\otimes(\alpha v_j)$.
I want to show that $f$ is an isomorphism between $V^*\bigotimes_\mathbb{K} W$ and $\text{Hom}_\mathbb{K}(V,W)$ by showing that $g$ and $f$ are inverse to each other, ie. that $g\circ f=\text{id}_{V^*\bigotimes_\mathbb{K} W}$ and $f\circ g=\text{id}_{\text{Hom}_\mathbb{K}(V,W)}$. I have been trying for several hours to compute these compositions but I don’t seem to come to a conclusion. Could somebody help me out with this problem? Thank you very much for your help!