On a finite-dimensional vector space over R or C, is every norm induced by an inner product? I know that this can fail for infinite-dimensional vector spaces. It just struck me that we never made a distinction between normed vector spaces and inner product spaces in my linear algebra course on finite-dimensional vector spaces. Why I actually care about it: I wonder why the unit sphere in a finite-dimensional normed vector space is weakly closed. Obviously the statement should be that a sequence in a finite-dimensional space converges weakly if and only if it converges strongly, but I'm not sure how to go about this without using an inner product.