Two closed subspace whose sum is not closed?

In summary, the conversation discusses finding an example of two closed subspaces in a normed or Banach space whose sum is not closed. The example given is in the Hilbert space \mathcal{H}=\ell^2(\mathbb{N}), where A={(x,0) : x in \mathcal{H}} and B={(x,Tx) : x in \mathcal{H}}. It is shown that the sum A+B is not closed due to the range of the bounded linear operator T being a proper dense subspace of \mathcal{H}. Additional resources are suggested for further exploration of the topic.
  • #1
quasar987
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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!
 
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  • #2
How about this: Consider the Hilbert space [itex]\mathcal{H}=\ell^2(\mathbb{N})[/itex] of square-summable sequences of reals. Let {en} be the standard o.n. basis for [itex]\mathcal{H}[/itex], and define T on [itex]\mathcal{H}[/itex] by letting T(en)=(1/n)*en and extending linearly. This is a bounded linear operator on [itex]\mathcal{H}[/itex]. Next, consider the space [itex]\mathcal{H} \oplus_2 \mathcal{H}[/itex], which is simply the direct sum of two copies of [itex]\mathcal{H}[/itex] given the 2-norm coordinate wise. (This is still a Hilbert space.) Let A={(x,0) : x in [itex]\mathcal{H}[/itex]} and B={(x,Tx) : x in [itex]\mathcal{H}[/itex]}. Then A and B are subspaces of [itex]\mathcal{H} \oplus_2 \mathcal{H}[/itex], and A+B is closed there iff {Tx : x in [itex]\mathcal{H}[/itex]} is closed in [itex]\mathcal{H}[/itex]. But the range of T is a proper dense subspace of [itex]\mathcal{H}[/itex]. Thus, A+B cannot be closed.
 
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  • #3
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.
 
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  • #4
Very nice! and congratulations on the fruitful google search ;)
 

Related to Two closed subspace whose sum is not closed?

1. What is a closed subspace?

A closed subspace is a subset of a vector space that contains all of its limit points. In other words, any sequence of points in the subspace that converges must also have its limit point in the subspace.

2. How do you define the sum of two closed subspaces?

The sum of two closed subspaces is defined as the set of all possible sums of vectors from each subspace. In other words, if A and B are two closed subspaces, then their sum is denoted as A + B and is defined as {a + b | a ∈ A, b ∈ B}.

3. Can the sum of two closed subspaces be closed?

Yes, it is possible for the sum of two closed subspaces to be closed. This happens when the two subspaces have a non-empty intersection, meaning there are vectors that belong to both subspaces. In this case, the sum of the subspaces will also be a closed subspace.

4. What are some examples of two closed subspaces whose sum is not closed?

One example is the subspaces A = {(x, y) | x ≥ 0, y = 0} and B = {(x, y) | x = 0, y ≥ 0} in R^2. The sum of these two subspaces is the x-axis, which is not a closed subspace. Another example is the subspaces A = {(x, y) | x ≥ 0, y = 0} and B = {(x, y) | x = 0, y > 0} in R^2. The sum of these two subspaces is the first quadrant, which is also not a closed subspace.

5. What implications does a sum of closed subspaces not being closed have in mathematics?

The fact that a sum of two closed subspaces can sometimes not be closed has important implications in functional analysis and topology. It means that in general, the sum of two closed subspaces is not always a closed subspace, and thus cannot be treated as such in mathematical proofs and calculations. This concept is also important in understanding the properties of vector spaces and their subsets.

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