Let X be a normed space. Let $v_1, v_2… v_n$ be vectors in X. When is $Span({v_1, v_2, … v_n})$ closed?
This question is motivated by a question which I had on a problem sheet (this wasn't the exact topic of the question but a required assumption for the proof), which said a span of finite set of vectors from a Hamel Basis of a Banach Space is closed. I'm not sure why that is and I'm not sure in general when the Span is closed, thanks!
Best Answer
It always is. That span, endowed with the restricted norm, is a finite-dimensional normed space, and therefore it's bi-Lipschitz isomorphic to $\Bbb R^k$ for some $k\le n$. Therefore the restricted metric is complete. A subspace of a metric space for which the restricted metric is complete is necessarily closed.