Two closed subspace whose sum is not closed?

[quasar987]

What would be an example of two closed subspaces of a normed (or Banach) space whose sum A+B = {a+b: a in A, b in B} is not closed?

I suppose we would have to look in infinite dimensional space to find our example, because this is hard to imagine in R^n!

 

[morphism]

How about this: Consider the Hilbert space $\mathcal{H}=\ell^2(\mathbb{N})$ of square-summable sequences of reals. Let {e_n} be the standard o.n. basis for $\mathcal{H}$, and define T on $\mathcal{H}$ by letting T(e_n)=(1/n)*e_n and extending linearly. This is a bounded linear operator on $\mathcal{H}$. Next, consider the space $\mathcal{H} \oplus_2 \mathcal{H}$, which is simply the direct sum of two copies of $\mathcal{H}$ given the 2-norm coordinate wise. (This is still a Hilbert space.) Let A={(x,0) : x in $\mathcal{H}$} and B={(x,Tx) : x in $\mathcal{H}$}. Then A and B are subspaces of $\mathcal{H} \oplus_2 \mathcal{H}$, and A+B is closed there iff {Tx : x in $\mathcal{H}$} is closed in $\mathcal{H}$. But the range of T is a proper dense subspace of $\mathcal{H}$. Thus, A+B cannot be closed.

 

[morphism]

I was googling to see if there's a better example, and I found the following http://www.hindawi.com/GetArticle.aspx?doi=10.1155/S0161171201005324. You might find it interesting.

Also, apparently this problem is discussed in the books A Hilbert Space Problem Book by Halmos and Elements of Operator Theory by Kubrusly. Try to see if your library has a copy of either.

 

[quasar987]

Very nice! and congratulations on the fruitful google search ;)