Tagged Questions
11
votes
1answer
229 views
Open Mapping Theorem: counterexample
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 ...
9
votes
2answers
174 views
If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?
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 ...
6
votes
1answer
324 views
An example of a norm which can't be generated by an inner product
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 ...
8
votes
3answers
334 views
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?
