Span of infinitely many vectors in a Banach Space

Question

Let $X$ be a Banach Space and $S$ be any infinite collection of linearly independent elements in $X$. There are examples that $Span(S)$ is not closed in $X$, like $S = \{x^n, n = 1,2,\cdots\}$ in $X = C([0,1])$ with the sup norm. I wonder if this is necessarily the case, i.e., $span(S)$ is necessarily not closed in $X$. If not, can we impose some assumption on $X$ to make $Span(S)$ necessarily not closed?

Any hint or comment is appreciated.

Answer 1

There is no Banach space of countably infinite Hamel dimension. This is an easy consequence of Baire Category Theorem. Hence, $span S$ cannot be closed.