Let V be a finite dimensional vector space over field F and let v belong to V,v not equal to 0.Then there exists some f in V* such that f(v) is not equal to 0.
(where V* is dual space of V)
f is a mapping , isn't it? It is termed a linear functional if it's a linear mapping from V into F.
If is a basis of Vover F, let be the linear functionals defined by Kronecker delta notation:
Then set is a basis for . Anf those linear functionals f_i are unique since they're defined on a basis of V.