TITLE: $L^p$ space question
QUESTION [5 upvotes]: Assume $(X,\mathcal{M},\mu)$ is a measure space and for some $1\leq p<\infty$, $1\leq q<\infty$, $L^p(\mu)\subset L^q(\mu)$.  Prove there is a constant $C>0$ so that $\|f\|_q\leq C\|f\|_p$ for all $f\in L^p(\mu)$.

I need help getting started on this.

REPLY [7 votes]: $L^p(\mu)$ and $L^q(\mu)$ are Banach spaces, so we can apply the closed graph theorem: if $f_n\to f$ in $L^p$ and $f_n\to g$ in $L^q$ then passing to a subsequence $f_{n_k}\to f$ and $g$ almost everywhere. Hence the graph of the identity $\iota\colon L^p\to L^q$ is closed, so $\iota$ is continuous (and linear).