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Math Help - Bounded linear functions on Lp spaces

  1. #1
    GTO
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    Bounded linear functions on Lp spaces

    Let g be an integrable functions on [0,1] and 1/p+1/q=1 with 1<p<\infty. Suppose that there is a constant M>0 such that |\int fg| \leq M||f||_p for all bounded measurable functions f. Prove that g \in L^q and ||g||_q \leq M.

    any help is appreciated please.
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    Quote Originally Posted by GTO View Post
    Let g be an integrable functions on [0,1] and 1/p+1/q=1 with 1<p<\infty. Suppose that there is a constant M>0 such that |\int fg| \leq M||f||_p for all bounded measurable functions f. Prove that g \in L^q and ||g||_q \leq M.
    Here's the classic proof as given in Dunford and Schwartz. First, we may assume that g(t)\geqslant0\; (0\leqslant t\leqslant1)<br />
. Reason: let \alpha (t) = \overline{g(t)}/|g(t)| (with \alpha(t)=1 if g(t)=0). In other words, \alpha(t) is the complex number of absolute value 1 such that \alpha(t)g(t)\geqslant0. Replace g by \alpha g, which has the same norm as g in any of the L^p spaces, and is always real and non-negative. We can then assume that
    \textstyle\int fg \leqslant M\|f\|_p\qquad ({\color{blue}^*})
    for all non-negative f\in L^p. (We are only told that this holds for bounded f, but you can easily see from the Monotone Convergence theorem that it holds for all f\in L^p.)

    Now comes the clever part. It follows by putting f\equiv1 in (*) that \|g\|_1\leqslant M. Also, g^{1/p}\in L^p, with \|g^{1/p}\|_p = \|g\|_1^{1/p} \leqslant M^{1/p}. Therefore, putting f=g^{1/p} in (*), we get \textstyle\int g^{1+\frac1p}  \leqslant M\|g^{1/p}\|_p \leqslant M^{1+\frac1p}.

    Now let g_2 = g^{1+\tfrac1p}. We have shown that g_2\in L^1, with \|g_2\|_1\leqslant M^{1+\tfrac1p}. So g_{2}^{1/p} \in L^p, and we can repeat the argument in the previous paragraph to get \textstyle\int g^{1+\tfrac1p + \tfrac1{p^2}}\leqslant M^{1+\tfrac1p + \tfrac1{p^2}}. Thus g_3 = g^{1+\tfrac1p + \tfrac1{p^2}} \in L^1, with \|g_3\|_1 \leqslant  M^{1+\tfrac1p + \tfrac1{p^2}}.

    Continue in that way to see that \textstyle\int g^{1+\tfrac1p + \ldots + \tfrac1{p^n}}\leqslant M^{1+\tfrac1p + \ldots + \tfrac1{p^n}} for n=1,2,3,... . But \sum_{n=0}^\infty1/p^n = \frac1{1-\tfrac1p} = q. By Fatou's lemma, we can let n\to\infty and get \textstyle\int g^q \leqslant M^q, from which g\in L^q with \|g\|_q \leqslant M.
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