spanning sets of dual space
refers to the kernel. Note that the domain need not be finite-dimensional.
The “only if” part is easy: if for some scalars , and is such that for all , then clearly too.
The “if” part will be proved by induction on .
Suppose . If , then the result is trivial. Otherwise, there exists such that . By hypothesis, we also have . Every can be decomposed into where , and is a scalar. Indeed, just set , and . Then we propose that