The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to ...
Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
I realize that every inner product defined on a vector space can give rise to a norm on that space. The converse, apparently is not true. I'd like an example of a norm which no inner product can ...
Is there an easy example of a vector space which can not be endowed with the structure of a Banach space
Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?