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