we know that $L^p$ is banach space for any $p\geq 1$.
My question: Is there any other banach space that is bigger than $L^p$?.
In fact, I have an exercice that I don't have any idea: prove that space $(E,||.||_E)$ is banach space with norm $$||f||_E = \sup_{p\geq 2} \frac{||f||_p}{\ln{p}}.$$
Could any one give me any idea for this?