Let $F$ be a left vector $K$-space and $X$ a set. Denote by $K_d$ be the canonical right vector $K$-space. There exists an injective mapping $$g:K^X_d\otimes F\rightarrow F^X$$ such that $g(\alpha\otimes y)=(\alpha_xy)_{x\in X}$. Let $(f_\lambda)_{\lambda\in L}$ be a basis of $F$. Take $v\in(K^X_d)^{(L)}$. Then $$g(\sum_{\lambda\in L}v_\lambda\otimes f_\lambda)=\left(\sum_{\lambda\in L}v_\lambda(x)f_\lambda\right)_{x\in X}.$$ I am having problems showing that this latter family has a finite support.
Write $S:=\{\lambda\in L\ |\ v_\lambda\ne0\}$ and $T:=\{x\in X\ |\ \sum_{\lambda\in L}v_\lambda(x)f_\lambda\ne0\}$. My idea is to show that $|T|\leq|S|$. This would require either an injection from $T$ to $S$ or a surjection from $S$ to $T$. I can't seem to construct either. Any suggestions?