Can someone provide me with a sketch of proof / hint for this exercise:
Let K $\subset$ $L^1([0,1]\,,\, \mu)$ be a closed linear subspace. If $\forall$ $f \in K$, $\exists \, p > 1$ such that $f \in L^p$, then $\exists \, p > 0 \Rightarrow\, K \subset L^p$.
I understand that the uniform boundedness principle should be applied here, but I have no clue how. Thanks for any help in advance.
