Suppose that $U$ is a normed vector space and $S\subset U$ is dense.prove that every element of $U $ can be written as an absolutely convergent series of the finite linear combination of the elements of $S $.
$x\in U$, because $S $ is dense, we can write $x-s_1 \in B_1 (0) $ (the unit ball centered at $0$), repeatedly $x-(s_1+\ldots +s_n)\in \cap_1^n B_{1/n} (s_{n-1})$.
But how to prove the statement? I mean the Absolute convergence.
Thanx a lot
Write an arbitrary $x\in U$ as the limit of a sequence $(x_n)_n$ in $S$ such that $|x_n-x|<\frac1{n^2}.$ Then $|x_n-x_{n-1}|<\frac2{n^2}$ and the series
$$x=\sum_{n=0}^\infty s_n=\sum_{n=0}^\infty(x_n-x_{n-1})$$
converges absolutely.