$\ell^p$ as a direct summand of $L^p$

Question

I've been struggling with the following problem from a previous year's quals, and I don't know where to look it up (or even if it's supposed to be too obvious to write down).

How do we embed $\ell^p$ as a direct summand of $L^p(0,1)$? In other words, how do we find an isomorphism $L^p(0,1)\cong \ell^p\oplus V$ of Banach spaces for some space $V$?

(I think it's clear how to get a stupid embedding...just pick countably many functions with disjoint support, but how do we show it is a direct summand?)

Answer 1

Let $(A_n)_{n\in\mathbb{N}}$ be the set of disjoint measurable subsets of $L_p(\Omega,\mu)$ that all of positive measure. Then consider bounded liner operator $$ P:L_p(\Omega,\mu)\to L_p(\Omega,\mu):f\mapsto \sum_{n=1}^\infty\left(\mu(A_n)^{-1}\int_{A_n}f(\omega)d\mu(\omega)\right)\chi_{A_n} $$ One can show that $\operatorname{Im}(P)\underset{{\mathbf{Ban}_1}}{\cong}\ell_p(\mathbb{N})$ and $P$ is a norm $1$ projector. Therefore $V=\operatorname{Ker}(P)$.