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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

    HallsofIvy

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    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.
     
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