Bounded linear functions on Lp spaces

**GTO** · December 26th 2009, 04:22 PM

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.

**Opalg** · December 27th 2009, 11:10 AM

Originally Posted by GTO

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 $\|g\|_q \leqslant M$ .

Thread: Bounded linear functions on Lp spaces

LinkBack

Thread Tools

Display

Bounded linear functions on Lp spaces

Similar Math Help Forum Discussions

Linear Spaces

vector spaces, linear independence and functions

Sum of Linear Spaces

Bounded spaces

Linear Spaces

Search Tags