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Two closed subspace whose sum is not closed?

  1. Feb 10, 2008 #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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  3. Feb 10, 2008 #2

    morphism

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    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.
     
    Last edited: Feb 10, 2008
  4. Feb 10, 2008 #3

    morphism

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    I was googling to see if there's a better example, and I found the following paper. 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.
     
    Last edited: Feb 10, 2008
  5. Feb 10, 2008 #4

    quasar987

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