optimization – Approximate a large linear span with a small one

The subject is outside my field of expertise (I’m not sure if the tags are correct – feedback is appreciated).

The claim below could be true or false. If false I’d like to replace it with its best true approximation. If true I’d like to prove it.

First of all, I’d like to know what I have to study to understand my own question.

Claim Let $Esubseteq{mathbb R}^N$. Let $Vert{cdot}Vert$ be the 1-norm. Then
forallvarepsilon exists n=n(varepsilon) exists e_1,dots,e_nin E forall finlangle Erangle exists ginlangle e_1,dots,e_nranglequad Vert f- gVert<varepsilon Vert fVert.

Note that $n$ does not depend on $N$ nor $E$.