What is an example of a closed subspace of a Banach space whose linear complement (direct sum decomposition) is not closed?
Note that "linear complement" is not unique. Anyway, if $L$ is a closed subspace of $X$ having a closed linear complement $M$ then $L$ is (topologically) complemented, i.e. there is a continuous linear projection onto $L$. This follows from the open mapping theorem: $L\times M\to X$, $(\ell,m)\mapsto \ell+m$ is a bijective continuous linear operator between Banach spaces hence its inverse is continuous.
An example of a closed subspace without closed complement is thus $c_0 \subseteq \ell^\infty$.