If $V$ is a subspace of $\ell^2$ that doesn't contain a sequence which has only finitely many nonzero elements, can the closure of $V$ contain it?

Question

Let $V$ be a infinite dimensionale subspace of $\ell^2$ such that it doesn't exist a sequence $v \in V$ which has only finitely many nonzero elements.

Let $W=\overline{V}$ be the closure of $V$

My question is: can $W$ space contain a sequence which has only finitely many nonzero elements ?

Thanks

Answer 1

Let $0<a<1$ and take $V=$ Span$(v_n)$ where $$v_n=(1, a^{2^n}, a^{2^{n+1}},\ldots)$$

Then $V$ satisfies your condition but $W$ contains $(1, 0, 0, \ldots)$.