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Show map is injective

  1. Mar 3, 2008 #1

    quasar987

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    [SOLVED] Show map is injective

    1. The problem statement, all variables and given/known data
    Going crazy over this.

    Let 1<p<2 and q>=2 be its conjugate exponent. I want to show that the map T: L^p(E) --> (L^q(E))*: x-->T(x) where

    [tex]<T(x),y> = \int_Ex(t)y(t)dt[/tex]

    is injective.

    This amount to showing that if

    [tex]\int_Ex(t)y(t)dt=0[/tex]

    for all q-integrable functions y(t), then x(t)=0 (alsmost everywhere)

    Should be easy but I've been at this for an hour and I don't see it!
     
    Last edited: Mar 3, 2008
  2. jcsd
  3. Mar 4, 2008 #2

    quasar987

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    Got it. Turns out that T is a linear isometry and every linear isometry is injective! (If T(y)=0, then ||T(y)|| = ||y|| = 0 ==> y=0).
     
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