Do there exist closed subspaces $X$, $Y$ of Banach space, such that $X+Y$ is not closed?

Question

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

Answer 1

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.

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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.