Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Normed and topological vector spaces

  1. Jul 4, 2009 #1
    Can someone explain how topological vector spaces are more general than normed ones. So this means that if I have a normed vector space, [tex]X[/tex], it would have to induce a topology. I was thinking the base for this topology would consist of sets of the form [tex]\{y\in X: ||x-y||<\epsilon, \textrm{for some x$x\in X$ and $\epsilon>0$}\}[/tex], which is the analogue of the open ball centred at y with radius epsilon. Is this correct?
  2. jcsd
  3. Jul 4, 2009 #2
    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 [tex]\mathbb{R}^\mathbb{N}[/tex] 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.
  4. Jul 4, 2009 #3


    User Avatar
    Science Advisor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook