I need help proving that statement A implies B:
STATEMENT A: $\exists!$ isomorphism $\Delta: V \to \mathcal{F}_{00}(S;\mathbb{K}) $ satisfying $\Delta e_s = \delta_s$ for all $s \in S$.
STATEMENT B: for any vector space $W$ over $\mathbb{K}$ and map $\alpha \in \mathcal{F}(S;W)$ there is a unique map $A \in L(V;W)$ s.t. $Ae_s=\alpha (s)$ for all $s \in S$.
NOTE: $(e_s)_s\in S$ is an indexed set in a vector space $V$ over $\mathbb{K}$ (real or complex field). $\mathcal{F}_{00}(S;\mathbb{K})$ denotes the set of maps from $S$ to $\mathbb{K}$ having finite support; whereas $\mathcal{F}(S;W)$ denotes the set of functions from $S$ to $W$.
Any hint, tip, etc will be appreciated. Thank you!