Norm of a linear functional of a subspace of $\ell^2$

Question

Let $S \subset \ell^2$ be a infinite linearly independent compact subset of $\ell^2$ over $\Bbb C$

Let $V=span(S)$ and let $T:\overline{V} \to \Bbb C$ be a linear functional of $\overline{V}$

I would like to know if exist $v \in V$ such that $\lVert T\rVert = |T(v)|$

Thanks for any suggestion