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Linear transformations between normed linear spaces

  1. Feb 21, 2007 #1
    Hi, ok I'm working with linear transformations between normed linear spaces (nls)

    if T :X -> Y nls's is a linear transformation, we define the norm of T, ||T||: sup{||T|| : ||x||<=1}

    I want to show that for X not = {0}
    ||T||: sup{||T|| : ||x|| = 1} frustratingly the books all assume that this step is obvious... I don't see how.

    Intuitively I can see that is true, using the fact that (I think) T is a bounded function, and we can manipulate things to make the whole function be contained within a closed unit sphere...
    Have always had a mental block with inf's and sups...don't know why...
  2. jcsd
  3. Feb 21, 2007 #2


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    Work with x/||x||. It will be obvious.
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