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

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

@AnneBauval Thanks! But could you explain why we can identify the complementary subspace with the quotient space? because my question is about quotient, but you provide the links about complementary subspaces. Can we say they are the same thing? — Hamilton, Jun 10 '23 at 03:28
@AnneBauval Thank you, I think I got it, when we write something like $V \cong U \oplus V/U$ as Banach space direct sum, it means we can find a subspace of $V$ such that is isomorphic to $V/U$, which means we can find a complementary subspace. — Hamilton, Jun 10 '23 at 05:01
More precisely: the existence of an isomorphism $V \cong U \oplus V/U$ (equal to the identity on $U$) i.e. of a complement of $U$ (not only a subspace of $V$) isomorphic to $V/U,$ is equivalent to the existence of a complement of $U$. Every complement is isomorphic to the quotient. — Anne Bauval, Jun 10 '23 at 05:41
@AnneBauval Thanks! I have one more confusion left, if we have established $V/U \cong W$ as Banach spaces, where $W$ some other Banach space, $U$ is a closed subspace of $V$, can we write directly $V \cong U \oplus W$? I believe for the same reason you gave me, the answer is no. — Hamilton, Jun 10 '23 at 13:45
No! Your conjecture (where the isomorphism $V \cong U \oplus W$ is supposed to be equal to the identity on $U$), applied to $W=V/U,$ would imply that every closed subspace $U$ of a Banach space $V$ is complemented... — Anne Bauval, Jun 10 '23 at 13:50
Thanks! I got it, we can't write directly $V \cong U \oplus W$. Thank you so much. — Hamilton, Jun 10 '23 at 14:03
Note however that if the closed subspace $U$ has a finite codimension, then it is complemented. — Anne Bauval, Jun 10 '23 at 14:12
An example Banach space: The Banach space $l^\infty$ and the subspace $c_0 \subset l^\infty$. There is no subspace of $l^\infty$ linearly homeomorphic to $l^\infty/c_0$. — GEdgar, Jun 12 '23 at 20:12

Hamilton · Accepted Answer · 2023-06-13T03:24:05.447

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.

Banach space direct sum

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