Is a linear span of finite set from a finite dimensional space topologically closed?

Question

Let $S=\{x_1,\ldots,x_m\} \subset \mathbb{C}^n $ is it true that:

$$ Span (S) = \overline{Span (S)} $$

Must we assume both of the following assumptions? or one of them will be enough?

• The spanning set, S, has a finite number of elements
• The vector space (e.g. $\mathbb{C}^n $) is of finite dimension

Answer 1

In a normed metric space $X$ (such as $\mathbb{C}$) all subspaces spanned by a a finite number of vectors are closed.

I do not know if this is also the case for arbitrary topological vector spaces.