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Math Help - Application of Closed Graph Theorem

  1. #1
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    Application of Closed Graph Theorem

    Hi all,

    Any help (hints included) with this is appreciated!

    Let  ( \alpha_{n} ) be a sequence of scalars such taht  \sum \alpha_{n} \beta_{n}<br />
converges whenever  ( \beta_{n} ) \in c_{0}<br />
. Show that there is a linear operator  T: c_{0} \rightarrow l^{1}<br />
given by

     T(\beta_{n}) = (\alpha_{n} , \beta_{n})<br />

    Show that  T has a closed graph. Use the Closed Graph Theorem to deduce that (\alpha_{n}) \in l^{1}<br />
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Mimi89 View Post
    Hi all,

    Any help (hints included) with this is appreciated!

    Let  ( \alpha_{n} ) be a sequence of scalars such taht  \sum \alpha_{n} \beta_{n}<br />
converges whenever  ( \beta_{n} ) \in c_{0}<br />
. Show that there is a linear operator  T: c_{0} \rightarrow l^{1}<br />
given by

     T(\beta_{n}) = (\alpha_{n} , \beta_{n})<br />

    Show that  T has a closed graph. Use the Closed Graph Theorem to deduce that (\alpha_{n}) \in l^{1}<br />
    This question doesn't make msuch sense. What is c_0? Is it \ell^\infty?
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  3. #3
    Junior Member
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     c_0 is the space of all sequences converging to 0. It is a subspace of  l^\infty .
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