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Thread: 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 $\displaystyle V \cap W = \{0\}$. Prove that $\displaystyle V\oplus W$ is closed $\displaystyle \iff V,W$ are both closed.

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

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

    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 $\displaystyle V \cap W = \{0\}$. Prove that $\displaystyle V\oplus W$ is closed $\displaystyle \iff V,W$ are both closed.

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

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

    I would really appreciate any help with this, thank you
    Are you aware of the result that under these circumstances $\displaystyle \displaystyle \pi_{V}$ and $\displaystyle \pi_{W}$ 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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