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Re: [harmonic] weak conv/ strong convergence

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  • ahmed ibrahim
    Dear All, Could anyone suggest a source where I can find the difference between weak and strong conversion. Thanks and Best Regards, Ahmed Ibrahim ...
    Message 1 of 4 , Mar 1, 2005
      Dear All,
      Could anyone suggest a source where I can find the
      difference between weak and strong conversion.

      Thanks and Best Regards,
      Ahmed Ibrahim

      --- h1l2star <h1l2star@...> wrote:

      >
      > Dear Members of the Harmonic Analysis
      >
      > I know that
      > [1] u(t)\in L^{00}(I,L^{2}(R^{n})), where
      > I\subset\R an interval.
      > and I want to show
      > [2] u(t)\in C(I,L^{2}(R^{2})).
      >
      > From [1] Eberlein-Smulian's Thm gives now a weak
      > convergent
      > subsequence \Lambda in L^{2}.
      > But I want to obtain strong convergence in L^{2} for
      > every sequence
      > t_{k}->t for k->00.
      >
      > I wanted to use Lebesgues Dominated Convergence
      > Theorem.
      > But therefore I need convergence almost everywhere.
      > And I think weak conv. doesn't imply conv. a.e. ?
      >
      > So I tried the following:
      > Because of weak conv. in L^{2} we can choose
      > \phi_{1},\phi_{2}\in C^{00}_{0}(R^{n})
      > such that
      > \int |(u(t_{k})-u(t))\phi_{1}|^{2}dx \leq \int
      > |u(t_{k})-u(t)|^{2}dx
      > \leq \int (u(t_{k})-u(t))\phi_{2}dx
      > and with k->00, k\in\Lambda we obtain the strong
      > conv.
      >
      > I used that the weak limit is equal to the strong
      > limit.
      > Is this always true or do I have to prove this in
      > every situation?
      >
      > I'm very grateful for answers, remarks and hints!
      >
      > With best regards
      >
      > h1l2star
      >
      >
      >
      >




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    • nir31415
      The notation L^{00}(I,L^{2}(R^{n})) is not clear. Please use standard notation, or better explain in words. Best, Nir. ... dx
      Message 2 of 4 , Mar 28, 2005
        The notation " L^{00}(I,L^{2}(R^{n})) " is not clear.

        Please use standard notation, or better explain in words.

        Best, Nir.




        --- In harmonicanalysis@yahoogroups.com, "h1l2star" <h1l2star@y...>
        wrote:
        >
        > Dear Members of the Harmonic Analysis
        >
        > I know that
        > [1] u(t)\in L^{00}(I,L^{2}(R^{n})), where I\subset\R an interval.
        > and I want to show
        > [2] u(t)\in C(I,L^{2}(R^{2})).
        >
        > From [1] Eberlein-Smulian's Thm gives now a weak convergent
        > subsequence \Lambda in L^{2}.
        > But I want to obtain strong convergence in L^{2} for every sequence
        > t_{k}->t for k->00.
        >
        > I wanted to use Lebesgues Dominated Convergence Theorem.
        > But therefore I need convergence almost everywhere.
        > And I think weak conv. doesn't imply conv. a.e. ?
        >
        > So I tried the following:
        > Because of weak conv. in L^{2} we can choose
        > \phi_{1},\phi_{2}\in C^{00}_{0}(R^{n})
        > such that
        > \int |(u(t_{k})-u(t))\phi_{1}|^{2}dx \leq \int |u(t_{k})-u(t)|^{2}
        dx
        > \leq \int (u(t_{k})-u(t))\phi_{2}dx
        > and with k->00, k\in\Lambda we obtain the strong conv.
        >
        > I used that the weak limit is equal to the strong limit.
        > Is this always true or do I have to prove this in every situation?
        >
        > I'm very grateful for answers, remarks and hints!
        >
        > With best regards
        >
        > h1l2star
      • nir31415
        Try these books 1. Functional Analysis: An Introduction / Yuli Eidelman, Vitali Milman and Antonis Tsolomitis 2. Introductory real analysis / A.N. Kolmogorov
        Message 3 of 4 , Mar 28, 2005
          Try these books

          1. Functional Analysis: An Introduction / Yuli Eidelman, Vitali
          Milman and Antonis Tsolomitis
          2. Introductory real analysis / A.N. Kolmogorov and S.V. Fomin

          But actually these notions are very simple, so try
          to search Google for them.

          Best, Nir.


          --- In harmonicanalysis@yahoogroups.com, ahmed ibrahim
          <asony11@y...> wrote:
          > Dear All,
          > Could anyone suggest a source where I can find the
          > difference between weak and strong conversion.
          >
          > Thanks and Best Regards,
          > Ahmed Ibrahim
          >
          > --- h1l2star <h1l2star@y...> wrote:
          >
          > >
          > > Dear Members of the Harmonic Analysis
          > >
          > > I know that
          > > [1] u(t)\in L^{00}(I,L^{2}(R^{n})), where
          > > I\subset\R an interval.
          > > and I want to show
          > > [2] u(t)\in C(I,L^{2}(R^{2})).
          > >
          > > From [1] Eberlein-Smulian's Thm gives now a weak
          > > convergent
          > > subsequence \Lambda in L^{2}.
          > > But I want to obtain strong convergence in L^{2} for
          > > every sequence
          > > t_{k}->t for k->00.
          > >
          > > I wanted to use Lebesgues Dominated Convergence
          > > Theorem.
          > > But therefore I need convergence almost everywhere.
          > > And I think weak conv. doesn't imply conv. a.e. ?
          > >
          > > So I tried the following:
          > > Because of weak conv. in L^{2} we can choose
          > > \phi_{1},\phi_{2}\in C^{00}_{0}(R^{n})
          > > such that
          > > \int |(u(t_{k})-u(t))\phi_{1}|^{2}dx \leq \int
          > > |u(t_{k})-u(t)|^{2}dx
          > > \leq \int (u(t_{k})-u(t))\phi_{2}dx
          > > and with k->00, k\in\Lambda we obtain the strong
          > > conv.
          > >
          > > I used that the weak limit is equal to the strong
          > > limit.
          > > Is this always true or do I have to prove this in
          > > every situation?
          > >
          > > I'm very grateful for answers, remarks and hints!
          > >
          > > With best regards
          > >
          > > h1l2star
          > >
          > >
          > >
          > >
          >
          >
          >
          >
          > __________________________________
          > Do you Yahoo!?
          > Yahoo! Mail - Find what you need with new enhanced search.
          > http://info.mail.yahoo.com/mail_250
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