Motivation: If $a$ and $b \ne 0$ are real numbers, then $a = b \cdot (a / b)$.
Question: Let $X$ be a Banach space and $M \subset X$ a closed subspace. Then, the quotient space $X / M$ is also a Banach space. Do we have $$ X = M \times (X / M) $$ in any sense?
(For example, "$\times$" could denote the product Banach space and "$=$" could mean "isomorphic".)
Such a subspace $M$ is called "complemented"
If $M$ is a closed subspace of $X$, and there exists another closed subspace $N$ such that $X = M \oplus N$, then $N$ is isomorphic to $X/M$. [Here, I mean that the map $M \oplus N \to X$ defined by $(m,n) \mapsto m+n$ is a homeomorphism from $M \oplus N$ onto $X$. The topology in $M \oplus N$ is the Cartesian product topology.]
Not all subspaces in a Banach space are complemented, but many common ones are. Of course in Hilbert space, every subspace is complemented. Also: finite-dimensional subspaces are complemented.
An example of a subspace that is not complemented: $c_0 \subset l^\infty$ is not compemented.
More difficult to prove, but true: if $X$ is a Banach space and every closed subspace is complemented, then $X$ is isomorphic to a Hilbert space.
A closed subspace $M$ that satisfies $X\approx M\times N$ for some other closed subspace $N$, is said to be complemented.
It is a fact that some Banach spaces have closed subspaces that are not complemented. See Example of a closed subspace of a Banach space which is not complemented?
