An introduction to functional analysis - download pdf or read online

By Mischa Cotlar; Roberto Cignoli

ISBN-10: 0444104402

ISBN-13: 9780444104403

ISBN-10: 0720420660

ISBN-13: 9780720420661

ISBN-10: 0720420954

ISBN-13: 9780720420951

This textbook emphasizes these subject matters correct and essential to the examine of research and likelihood thought. the 1st 5 chapters take care of summary size and integration. bankruptcy 6, on differentiation, incorporates a therapy of alterations of variables in Rd

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Example text

3 ⇒ 1. On a n(Lk x) ≤ λk n(x) → 0 lorsque k → ∞. 5 Endomorphismes dilatants Les endomorphismes dilatants sont caract´eris´es par le th´eor`eme suivant : Th´ eor` eme 21. Pour un endomorphisme bijectif et continu L d’un espace de Banach E il y a equivalence entre 1. Pour tout x ∈ E, limk→∞ L−k (x) = 0, 2. Il existe Λ > 1 tel que |λ| ≥ Λ pour toute valeur spectrale λ ∈ Spec (L), 3. Il existe une norme n sur E ´equivalente a ` la norme de E et un scalaire Λ > 1, tels que, pour tout x ∈ E, on ait n(Lx) ≥ Λn(x).

On proc`ede de mˆeme avec (L|Ed )−1 et l’on obtient : (L|Ed )−1 ≤ λ < 1. de L’application H est lipschitzienne et v´erifie Lip(H) ≤ Lip(k). Cela provient H(g) − H(g ′ ) = sup kg(L + k ′ )−1 (y) − kg ′ (L + k ′ )−1 (y) y∈E ≤ sup kg(x) − kg ′ (x) ≤ Lip(k) sup g(x) − g ′ (x) = Lip(k) g − g ′ . x∈E x∈E On peut maintenant utiliser la Proposition 31 dont nous venons de v´erifier les hypoth`eses (notons que la condition δ ≤ r(1 − λ − ǫ) est automatiquement satisfaite puisque l’on a ici r = ∞). On a donc prouv´e l’existence d’une unique 32 2 Points fixes fonction g ∈ Cb (E) telle que (L + k)(id + g) = (id + g)(L + k ′ ).

5 Le cas non lin´eaire : le th´eor`eme de Grobman-Hartman 29 ¯r , sa norme est major´ee par Alors f a un unique point fixe xf contenu dans B xf ≤ f (0) 1−λ−ǫ et, pour deux fonctions f et g perturbations de L v´erifiant les hypoth`eses cidessus, on a d(f, g) xf − xg ≤ 1−λ−ǫ ¯r → E o` u d est la distance associ´ee ` a la convergence uniforme des fonctions B c’est-` a-dire, d(f, g) = sup f (x) − g(x) . ¯r x∈B Preuve Notons fc = pc ◦ f et fd = pd ◦ f les projections de f sur Ec et Ed et ¯r → E par xd = pd (x).

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An introduction to functional analysis by Mischa Cotlar; Roberto Cignoli


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