Analyse fondamentale : espaces métriques, topologiques et by Szymon Dolecki

By Szymon Dolecki

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Additional resources for Analyse fondamentale : espaces métriques, topologiques et normés

Example text

Soit dj une métrique sur X j , bornée par 1, pour tout j G J. Montrer que (a) la fonction est une métrique. (b) si x = limn_>ooXn, alors il existe no et j G J tel que x n G Xj pour tout n > no. (c) la partie Xj de X est ouverte et fermée pour tout j G J. (d) Observer que si X j := { x j } , alors (X, d) est un espace discret. (27) * Soit Xy Y espaces métriques et / : X —>Y . ) est fermée pour tout fermé F de Y . (28) ★ Une application / : X -* Y est ouverte si f ( 0 ) est ouvert pour tout ouvert O de X .

20) i w ^ N i o o ^ i w ^ < 11^ . donc pour tous x , y y ^ di( xyy) < doo(x,y) < d2(xyy) < d\{xyy). (6) Soit (Xugi), (X 2yg2)> • • •, (Xnygn) des espaces métriques et D2{xyy) := Ç ^ k=l9k(xk>yk)2Y > D\{xyy) := $ ^* =10fe(**,îte), Doo(xyy) := maxi 0.

Toute isométrie est un homéomorphisme uniforme. D é m o n s t r a t i o n . Une isométrie est évidemment lipschitzienne, donc continue uniformément. 14) , f ( x o) = f ( x i) implique xo = aq, et l’application réciproque est également isométrie. □6 6. Pour tout n G N, soit f n(x) := x n. Les fonctions fo et f i sont lipschitzienne (avec les constantes 0 et 1 respectivement), tandis que f n sont continues mais pas uniformément continues pour n > 2. II. ESPACES MÉTRIQUES 33 4. P ro d u its dénom brables des espaces m étriques Rappelons que si J et X j , pour tout j G J , sont des ensembles, alors la projection : YljeJ X j Xk es^ définie par tTk(x):=x(k).

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