Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$
Prove $||f||_p\leq 1$.
How to prove? Thanks.
I have tried but failed: Let $F_f: L^q[0,1]\to\mathbb{R}$, $F_f(g)=\int_0^1 fg $. If for all $g\in L^q[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q, $$ then $$||f||_p=||F_f||=\sup_{||g||_q\neq 0}\frac{|F_f(g)|}{||g||_q}\leq 1.$$ Hence we need to prove for all $g\in L^q[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Let $\{\phi_n\}$ be a sequence of step functions such that $\lim_{n\to\infty}||\phi_n-g||_q=0$. I want to prove $\lim_{n\to\infty}|\int_0^1 f\phi_n d\mu|=\int_0^1 fg d\mu$. By holder inequality, $$ \int_0^1 f(\phi_n-g) d\mu\leq ||f||_p||\phi_n-g||_q\to ||f||_p\cdot 0. $$ But $f$ is not assumed to be in $L^p[0,1]$... what should I do? Thanks.
Your idea is good. To show $f\in L^p$, it suffices to show that the functional on $L^q$ defined by $F_f(g) := \int fg$ is continuous and linear. Linearity is clear, to show continuity, first let us look at the inequality for step function $g$ $$\left| \int fg\right| \leq ||g||_q$$ observe that this implies $$\left| \int f|g|\right| \leq ||g||_q$$ and further more by spiting $f$ into non-negative and negative parts, we have $$\left| \int |f||g|\right| \leq 2||g||_q $$ Now for any $g\in L^q$, let $\phi_n$ be a sequence of step functions such that $\phi_n \rightarrow g$ in $L^q$ and pointwise a.e., by Fatou's lemma, we can change the limit in the following way $$\left| \int fg \right| \leq \int|fg| = \int \lim_n |f\phi_n| \leq \liminf_n \int|f\phi_n| \leq \liminf_n 2||\phi_n||_q \leq 2||g||_q$$ which means that the functional $F_f$ is continuous.