Banach space direct sum

Question

This comes from an exercise in the book linear algebra done right.

Exercise 3.E.12: Suppose $U$ is a subspace of $V$ such that $V/U$ is finite-dimensional. Prove that V is isomorphic to $U \times V/U$.

My question is

1. can we generalize to infinite dimensional cases? (without imposing any topology on it, just with the algebraic method)

2. when we are dealing with Banach spaces. Given a Banach space $V$, $U$ being a closed subspace, can we write directly $V \cong U \oplus V/U$ as Banach spaces isomorphism, without any other additional assumptions?

Answer 1

The answer to the first question is "yes", as stated in the comment section of this question. The answer to the second question is "no".

Given a vector space $Z$, we have direct sum decomposition into two subspaces $Z = X \oplus Y$. More generally, given two vector spaces $X, Y$, the algebraic direct sum $X \oplus Y$ is the vector space of all ordered pairs $(x, y), x \in X, y \in Y$, with the vector operations defined coordinatewise. The spaces $X$ and $Y$ are algebraically isomorphic to the subspaces $\{(x, 0): x \in X\}$ and $\{(0, y): y \in Y\}$ of $X \oplus Y$, respectively.

Let $(X, \| \cdot \|_X)$ and $(Y, \| \cdot \|_Y)$ be normed spaces. The algebraic direct sum $X \oplus Y$ of $X$ and $Y$ becomes a normed space, called the topological direct sum of $X$ and $Y$, when it is endowed with the norm $\|(x, y)\| := \| x \|_X + \| y \|_Y$. The spaces $X$ and $Y$ are isometric to the subspaces $\{(x, 0) : x \in X\}$ and $\{(0, y): y \in Y\}$ of $X \oplus Y$, respectively.

Then it is easy to check that the quotient space $(X \oplus Y)/X$ is isomorphic to $Y$ and $(X \oplus Y)/Y$ is isomorphic to $X$. However, if $Y$ is a closed subspace of $X$, then $X$ may not be isomorphic $Y \oplus (X/Y)$, as the following example shows:

There is a separable Banach space $X$ that is not isomorphic to a Hilbert space and has a closed subspace $Y$ such that both $Y$ and $X/Y$ are isomorphic to Hilbert spaces([1]). If $X \cong Y \oplus X/Y$ as Banach spaces, then $X$ would be isomorphic to a Hilbert space, a contradiction.

[1] Enflo, Per; Lindenstrauss, Joram; Pisier, Gilles, On the ’three space problem’, Math. Scand. 36, 199-210 (1975). ZBL0314.46015.