Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

I am looking for an example of two closed subspaces of a Banach space, such that their sum is not closed.

share|cite|improve this question
up vote 17 down vote accepted

Simple examples can be obtained as follows: Let $E$ and $F$ be Banach spaces and suppose $T: E \to F$ is a bounded linear operator with non-closed range. Then $X = E \oplus 0$ and $Y = \operatorname{Graph}(T) = \{(e,Te)\,:\,e \in E\}$ are closed subspaces of $Z = E \oplus F$ with $X + Y$ not closed: If $Te_n \to f \in F \smallsetminus T(E)$ then $(0,Te_n) \in X + Y$ but $(0,f)$ isn't.

For an explicit example, take $E = \ell^1$, $F = \ell^2$ and $T: E \to F$ the obvious inclusion.

Added: It is not too hard to check that for $X$ finite-dimensional or of finite co-dimension the space $X + Y$ is always closed.

share|cite|improve this answer
Thank you Theo! – Jan Jun 3 '11 at 17:53
@Jan: You're welcome. I added a little bit of information to my answer. – t.b. Jun 3 '11 at 18:06
Very nice!! :-) – André Caldas Apr 28 '12 at 2:26

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.