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Homework Help: What does this notation mean?

  1. Oct 2, 2011 #1
    If you look here http://planetmath.org/encyclopedia/RiezsLemma.html [Broken], there seems to be something missing - nothing is said about the norm of (x_alpha) or about the norm of (s - x_alpha).

    Now, the same thing seems to happen here http://planetmath.org/encyclopedia/CompactnessOfClosedUnitBallInNormedSpaces.html [Broken], so I guess there's something about the notation that I'm not getting, rather than there being something actually missing.

    Can anyone help? To be honest, I fail to see how "lx_alphal and ls-x_alphal for every s in S" could be a statement.

    NB: The notation l.l is used to denote norm on the quoted webpages, rather than the more usual ll.ll
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Oct 2, 2011 #2
    (Riesz Lemma). Fix 0 < [itex]\alpha[/itex] < 1. If S[itex]\subset[/itex] E is a proper closed subspace of a
    Banach space E then one can find x[itex]_{\alpha}[/itex] [itex]\in[/itex] X with ||x[itex]_{\alpha}[/itex]|| = 1 and |s - x| [itex]\geq[/itex]  [itex]\alpha[/itex], for all s [itex]\in[/itex] S
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