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Topological vector spaces

  1. Feb 12, 2009 #1
    Let [tex]C([0,1])[/tex] be the collection of all complex-valued continuous functions on [tex][0,1][/tex].
    Define [tex]d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx[/tex] for all [tex]f,g \in C([0,1])[/tex]
    [tex]C([0,1])[/tex] is an invariant metric space.
    Prove that [tex]C([0,1])[/tex] is a topological vector space
  2. jcsd
  3. Feb 12, 2009 #2
    No thanks. Why don't you prove it for us?
  4. Feb 13, 2009 #3


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    If you have already proved that C([0, 1]) is a metrix space, and it is clearly a vector space, all you need to prove is that the functions F(f, g)= f+ g and F(a, f)= af are continuous.
  5. Feb 13, 2009 #4
    I don't know how to show F(a,f)=af is continuous, can I get some hints please.
  6. Feb 13, 2009 #5


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    Have you tried using the definitions? Or tried anything at all?

    (P.S. you should always try to use the definitions when you're stuck on any problem)
  7. Feb 13, 2009 #6
    F is continuous if for every open subset U of Y, F^{-1}(U) is open in X.
    So I let U be an open subset in Y, and let (a,f) be in F^{-1}(U). Then F(a,f) is in U. then I am stuck...
  8. Feb 14, 2009 #7
    Since you've got a metric space, it might be easier to use the epsilon-delta formulation of continuity to show it.
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