- 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

>

>

>

>

Do you Yahoo!?

Yahoo! Mail - Find what you need with new enhanced search.

http://info.mail.yahoo.com/mail_250 - 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:>

dx

> 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}

> \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 - 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