I know that every Hilbert space can be decomposed as the direct sum of two non-trivial closed subspaces, eg. taking the kernel and range of any non-trivial bounded projection. But I don't know what happens in Banach spaces.

Does there exist a Banach space $B$ with no complemented closed subspaces? In other words, such that there do not exist closed subspaces $U,V\subset B$ with $B=U\oplus V$?

  • $\begingroup$ mathworld.wolfram.com/ComplementedSubspace.html has an example. $\endgroup$ – R.N Nov 7 '15 at 7:54
  • $\begingroup$ math.stackexchange.com/questions/108284/… $\endgroup$ – R.N Nov 7 '15 at 7:55
  • $\begingroup$ Newman's paper from 1960 $\endgroup$ – A.Γ. Nov 7 '15 at 8:13
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    $\begingroup$ Aren't these examples of a specific closed subspace which is not complemented? I want an example of a space for which every closed subspace is not complemented. $\endgroup$ – Milind Nov 7 '15 at 8:52
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    $\begingroup$ Pick a nontrivial functional $\ell \in X^*$ and some $v\in X$ so that $\ell(v)=1$. Then $X = \ker \ell \oplus \langle v\rangle.$ $\endgroup$ – user99914 Nov 7 '15 at 9:16

Finite-dimensional subspaces are always complemented; this follows from the Hahn–Banach theorem.

Proof. Let $F$ be a finite-dimensional subspace of a Banach space $X$ and let $\{x_1, \ldots, x_n\}$ be a basis for $F$. Choose $f_1, \ldots, f_n\in F^*$ to be the coordinate functionals for the basis $\{x_1, \ldots, x_n\}$. Use the Hahn–Banach theorem to extend $f_j$ to bounded linear functionals on $X$; call any such extension $f_j$ too ($j\leqslant n$). Set

$$Px = \sum_{j=1}^n \langle f_j, x\rangle x_j\quad (x\in X).$$

Then $P$ is a projection with ${\rm im}\, P=F$. ${ \rm \square}$

The non-trivial variant of the question is really about the possibility of non-trivial decompositions into subspaces which are both infinite-dimensional (the spaces which lack this property exist and are called indecomposable Banach spaces). There is a whole industry concerning such spaces and there exist even spaces with a stronger property, i.e., there are spaces whose all infinite-dimensional suspaces are indecomposale (the so-called hereditarily indecomposable spaces).

There are examples of compact, Hausdorff spaces $K$ for which the Banach space $C(K)$ is indecomposale. (Of course such spaces cannot be hereditarily indecomposale as $C(K)$ always contains a copy of $c_0$ unless $K$ is finite.)

This is a nice introduction to hereditarily indecomposale spaces.

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