Math Help - Bounded linear functions on Lp spaces

1. Bounded linear functions on Lp spaces

Let g be an integrable functions on [0,1] and $1/p+1/q=1$ with $1. 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 $||g||_q \leq M$.

2. Originally Posted by GTO
Let g be an integrable functions on [0,1] and $1/p+1/q=1$ with $1. 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)
$
. 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 $\|g\|_q \leqslant M$.