Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

let $(X,|| || )$ be a norm linear space. And $M$ be a closed subspace of norm linear space .does there exist a closed subspace $N$ such that $X=M \oplus N $ . I know such an subspace $N$ exist .but i am not conform about such an $N$ is closed or not .

share|cite|improve this question
up vote 8 down vote accepted

In short, the answer is no in general. A well-known counterexample is given by $c_0$ in $\ell^\infty$. This was first proved by Phillips in 1940. There is a long history behind that, known as the complementary subspace problem for a Banach space $X$.

In 1971, Lindenstrauss and Tzafriri proved that given a Banach space $X$, every closed subspace is topologically complemented in $X$ if and only if $X$ is isomorphic to a Hilbert space.

Here are some details, to begin with a clarification of the notion of complement in a normed vector space, and in particular in a Banach space.

1) If two subspaces $M,N$, not necessarily closed, satisfy $X=M\oplus N$, we say that $M$ is algebraically complemented by $N$ in $X$. A stronger notion is that of topologically complemented. But these coincide in Banach spaces for pairs of closed subspaces.

If $X=M\oplus N$ algebraically, with $M,N$ subspaces of $X$, we have an isomorphism $T:M\times N\longrightarrow X$ by $T(x,y)=x+y$. Putting, say, the norm $\|(x,y)\|:=\|x\|+\|y\|$ on $M\times N$, and denoting by $P:x+y\longmapsto x$ the projection onto $M$ parallel to $N$, the following are equivalent:

  1. $T$ is a homeomorphism
  2. both $M$ and $N$ are closed and $P$ is bounded.

In this case, we say that $M$ is topologically complemented by $N$ in $X$.

If $X$ is a Banach space, and if $M,N$ are two closed subspaces of $X$, the closed graph theorem yields: $X=M\oplus N$ algebraically if and only if topologically.

2) So when $X$ is a Banach space, you are asking whether every closed subspace is topologically complemented. This is of course true when $X$ is a Hilbert space as it suffices to take $N=M^\perp$. Therefore it is true if $X$ is isomorphic to a Hilbert space. It is also true in a normed vector space with the additional assumption that $M$ has finite dimension of finite codimension.

A much more difficult result which appeared in 1971 after a long history of partial results is due to Lindenstrauss and Tzafriri: every infinite-dimensional Banach space which is not isomorphic to a Hilbert space contains a closed subspace which is not topologically complemented.

share|cite|improve this answer
Sorry for my error. Posting via smartphone is a humbling experience. – roo Jul 7 '13 at 23:53
@Kyle You'll find a detailed treatment of the fact that c0 is not complemented in linfty in Albiac-Kalton, Topics in Banach Space Theory, GTM. Together with many other things. – 1015 Jul 8 '13 at 0:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.