Let $l^p$ be the standard sequence space indexed by $\mathbb N$. I've heard it claimed that $l^p$ embeds into $L^p(0,1)$ in such a way that $$L^p(0,1)=l^p\oplus S$$ for some closed subspace $S\subset L^p$. Why is this?
For $p=2$ this is trivial, but the Hilbert space case gives no insight into the general case, it seems.
I thought about sending $e_n$ to $\chi_{[1/n,1/(n+1)]}$ (appropriately rescaled), but I don't think the image of this map is closed.
Certainly the image of that map is closed. It's the space of all $L^p$ functions which are constant on each one of those intervals. (Or: The image is closed because that map is an isometry.)
And the image is complemented: a natural choice for $S$ is the space of all $L^p$ functions $f$ such that the integral of $f$ over each one of those intervals is $0$.