Let $f$ be a function such that $\|fg\|_1<\infty$ whenever $\|g\|_2<\infty$. I would like to show that $\|f\|_2<\infty$. It seems that I should use some kind of Hölder inequalities, since we have $\|fg\|_1\leq \|f\|_2\|g\|_2$, but I don't know how. Any help would be appreciated. Thanks!
On some inequalities in $L_p$ spaces
1
$\begingroup$
real-analysis
functional-analysis
2 Answers
1
You have to assume that
$M := \sup \{ \|f \cdot g\|_1; \|g\|_2 \leq 1\}<\infty$
... otherwise it won't work. (Assume $M=\infty$. Then for all $n \in \mathbb{N}$ there exists $g_n \in L^2$, $\|g_n\|_2 \leq 1$, such that $\|f \cdot g_n\|_1 \geq n$. And this means that there cannot exist a constant $c$ such that $\|f \cdot g\|_1 \leq c \cdot \|g\|_2$, in particular $f \notin L^2$ (by Hölder inequality).)
-
0Now I see what you mean. I remember having seen this formula before: $\|f\|_2 = sup \{\|fg\|_1 ; \|g\|_2 = 1\}$. Then $f\in L^2$ if and only if the $sup$ is finite. – 2012-12-09
0
Do you have Hilbert space theory? Because then you can use that $T(g) = \int fg =
-
0Thanks! Unfortunately, I am not familiar with Hilbert space theory, so I don't really understand. – 2012-12-09