$L^\infty(\Omega)$ space

Question

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain.

Let $f \in L^p(\Omega)$ for all $p$.

Is it true that $f \in L^\infty(\Omega)$?

See the following question http://math.stackexchange.com/questions/66029/lp-and-lq-space-inclusion and the wiki link posted http://en.wikipedia.org/wiki/Lp_space#Embeddings — Chris Cave, Jul 18 '14 at 09:57

score 3 · Answer 1 · answered Jul 18 '14 at 09:43

3

Hint: $\Omega=(0,1)$, $f(x)=\log x$.

answered Jul 18 '14 at 09:43

Julián Aguirre

1 Answers1