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Math Help - Closed Direct Sum

  1. #1
    Senior Member slevvio's Avatar
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    Closed Direct Sum

    Hello all, I am having trouble with this problem and was wondering if anyone can help. I will post my partial proof and put the questions in bold, thanks very much!

    Let H be a Hilbert space and let V,W be linear subspaces such that V \cap W = \{0\}. Prove that V\oplus W is closed \iff V,W are both closed.

    Proof
    ( \impliedby) Let V,W both be closed. Let x_n be a convergent sequence in V \oplus W with x_n \rightarrow L \in H. We have x_n = v_n + w_n where v_n is a sequence in V and w_n is a sequence in W. Now if v_n and w_n both converge then v_n \rightarrow L_v and w_n \rightarrow L_w, where  L_v + L_w = L, but what if v_n and w_n are divergent? Is it not possible they diverge but both somehow converge to L when you add them together?

    ( \implies) Let V \oplus W be closed. Let v_n be a sequence in V \oplus W. Then v_n = v_n + 0 \in V \oplus W, so v_n \rightarrow L \in V \oplus W. How do I show that L \in V ?

    I would really appreciate any help with this, thank you
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by slevvio View Post
    Hello all, I am having trouble with this problem and was wondering if anyone can help. I will post my partial proof and put the questions in bold, thanks very much!

    Let H be a Hilbert space and let V,W be linear subspaces such that V \cap W = \{0\}. Prove that V\oplus W is closed \iff V,W are both closed.

    Proof
    ( \impliedby) Let V,W both be closed. Let x_n be a convergent sequence in V \oplus W with x_n \rightarrow L \in H. We have x_n = v_n + w_n where v_n is a sequence in V and w_n is a sequence in W. Now if v_n and w_n both converge then v_n \rightarrow L_v and w_n \rightarrow L_w, where  L_v + L_w = L, but what if v_n and w_n are divergent? Is it not possible they diverge but both somehow converge to L when you add them together?

    ( \implies) Let V \oplus W be closed. Let v_n be a sequence in V \oplus W. Then v_n = v_n + 0 \in V \oplus W, so v_n \rightarrow L \in V \oplus W. How do I show that L \in V ?

    I would really appreciate any help with this, thank you
    Are you aware of the result that under these circumstances \displaystyle \pi_{V} and \pi_{W} are continuous? This is a result of the closed graph theorem.
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  3. #3
    Senior Member slevvio's Avatar
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    Ah thanks I was not, but I cannot use that formula to solve this if it was on the exam! i will investigate that theorem though
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