Show that the vector space $V$ over Field $R$ with a basis $B$ and vector space $\text{Map}[B,R]$ are isomorphic.

Question

Question:

Show that the vector space $V$ over Field $R$ with a basis $B$ and vector space $\text{Map}[B,R]$ are isomorphic.

Show $V \cong \text{Map}[B,R]$.

($\;\text{Map}[B,R] := \{f \in \text{Map}[B,R] : f(a) = 0 \;\,\text{up to many finite}\;\, a \in B\}\;$.)

I understand so far that in order for two vector spaces to be isomorphic, they have to have the same $\dim(V ) =n.$

What I can’t get through in the above question is how to show that $$\dim(\text{Map}[B,R])= n .$$

Thanks alot