# Normed and topological vector spaces

1. Jul 4, 2009

### logarithmic

Can someone explain how topological vector spaces are more general than normed ones. So this means that if I have a normed vector space, $$X$$, it would have to induce a topology. I was thinking the base for this topology would consist of sets of the form $$\{y\in X: ||x-y||<\epsilon, \textrm{for some xx\in X and \epsilon>0}\}$$, which is the analogue of the open ball centred at y with radius epsilon. Is this correct?

2. Jul 4, 2009

### g_edgar

Yes, given a normed space, your definition is the standard way to define a topology, and the result is a topological vector space.

Now consider the set $$\mathbb{R}^\mathbb{N}$$ of all sequences of real numbers. With the obvious vector operations and the topology of pointwise convergence this becomes a topological vector space. But it can be shown that there is no norm for this space that produces this topology.

3. Jul 4, 2009

### HallsofIvy

Staff Emeritus
On the other hand, consider any vector space, V, with the topology in which the only open sets are V itself and the empty set. This is a topological vector space which cannot be derived from any metric, much less a norm.