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Norms in vector spaces

  1. Jan 17, 2010 #1
    Hi guys

    The norm is a continuous function on its vector space, but I a little unsure of how to interpret this. Does it mean that if we are in e.g. a Hilbert space with an orthonormal basis ei (i is a positive integer), then we have

    [tex]
    \left\| {\mathop {\lim }\limits_{N \to \infty } \sum\limits_{i = 1}^N {x_i e_i } } \right\| =\mathop {\lim }\limits_{N \to \infty } \left\| {\sum\limits_{i = 1}^N {x_i e_i } } \right\|
    [/tex]

    for some vector x = Σ xiei in the Hilbert space? Or does the above operation come from the continuity of taking limits?
     
  2. jcsd
  3. Jan 17, 2010 #2

    Landau

    User Avatar
    Science Advisor

    Yes, it does.
    I'm not sure what you mean by this. Maybe this will clear things up:

    You know that for a continuous function f and a converging sequence x_n, we have
    [tex]\lim_{n\to\infty}f(x_n)=f(\lim_{n\to\infty}x_n)[/tex],
    i.e. you can "pull the limit out of the argument of a continous function".

    Now, let f be the norm-function: [tex]f=\|...\|[/tex]. Then the above becomes
    [tex]\lim_{n\to\infty}\|x_n\|=\|\lim_{n\to\infty}x_n\|[/tex]
    i.e. you can "pull the limit out of the (argument of) the norm", which is what you did.
     
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