New PDF release: Analyse fonctionnelle : une introduction pour physiciens

By N. Boccara.

ISBN-10: 2729896058

ISBN-13: 9782729896058

Dans ce chapitre sont construits les outils indispensables a l'élaboration des théories qui seront développées par los angeles suite. l. a. concept de mesure у joue un rôle essentiel. Les résultats les plus importants sont le théorème de l. a. convergence dominée de Lebesgue et le théorème de Fubini relatif a l'interversion des ordres d’intégration dans les intégrales multiples.
Plus encore dans ce chapitre que dans les suivants, il est recommande de se familiariser avec ces théorèmes, en étudiant les différents exemples et en essayant de faire les exercices, avant d'en aborder l. a. démonstration.

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Extra info for Analyse fonctionnelle : une introduction pour physiciens

Example text

We call functions G, S -continuous, if they are realvalued continuous functions on R such that,for the position operator Q, GQ : S ! S , is continuous in the standard countably normed topology on the Schwartz space S . Theorem 4. If the force F is S -continuous, then given " 0, Heisenberg's equations of motion approximate Newton's equations of motion to within " on each member of a collection of open sets fW r; "g of state space ES , indexed by the standard real numbers r and " 0. That is, on each fW r; "g, ,  13 F Q W , F Q^ W ": Proof.

In this case, the eigenspaces are one-dimensional and are spanned by the characters of G. Research supported by the Australian Research Council ARC. 30 VILENKIN BASES IN NON-COMMUTATIVE Lp -SPACES 31 2. Preliminaries There is an extensive literature concerning harmonic analysis on compact Vilenkin groups that is related to the theme of present article. In particular we refer the reader to AVDR , BaR , Gos , SWS , Sc , Si1,2 , Vi , W , Wa , Y and the references contained therein. We be concerned here with Vilenkin groups of the form Gm = Q1 shall Z m k , equipped with the product topology and normalized Haar k measure.

The nal summand is also because by assumption the function F : R ! R is continuous everywhere in the usual topology on R . Because given " 0, there exists a xa  0, such that F x , F xa ", whenever jx , xa j . Apply this to x = tr Q and xa = tr r Q. Therefore, given r , for any in N  r ; Q;  N  r ; F Q; ", the inequality 16 holds. The question remains whether we can construct su ciently many of these open sets. for a given smooth function F , the family  In general, of open sets W r; " , does not form an open cover of state space ES .

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Analyse fonctionnelle : une introduction pour physiciens by N. Boccara.


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