How to prove that the $L^p$ spaces are infinite dimensional It is well-known that (given a measure space $(S,\mathcal A,\mu)$ and $1\le p\le\infty$) the Banach space $L^p(S,\mathcal A,\mu)$ has infinite dimension.
Is there an easy way to proof this statement (or a suitable reference (preferably a book) where I can find this result)?

Since $l^{p}$ embeds isomorphically into $L^{p}$,and it's easy to check that $l^{p}$ is infinitive dimensional.
