Closed with respect to norm in normed spaces In general, it is true that finite dimensional normed vector subspaces are closed. Does this mean they are closed with respect to any norm? 

Your question is in some sense not well-posed: to be closed for a vector subspace $U$ of a normed space $(V,\|\cdot\|)$ is a topological property, where one considers the topology on $V$ induced by the norm on $V$. However if $U$ has finite dimension, then all norms on $U$ induce the same topology. Since the restriction $\|\cdot\||_U$ is a norm the answer to your question in some sense is Yes.
