New PDF release: Cours d'analyse : Vol. 1

By Laurent Schwartz

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Extra info for Cours d'analyse : Vol. 1

Example text

Et ces : "Il existe . . ' sont souvent accompagnés de certaines restrictions. On note entre ces restrictions. parenthèses ( ) Supposons que nous voulions exprimer, pour une fonction réelle # d'une variable réelle, la propriété d'être continue en tout point. Dire qu'elle est continue en un point a, c'est dire que, quel que soit E 7 0 , il existe y ) 0 tel que 1% - 4 < '1 entrafne I$)(z)-t@,I 4 & ; autrement dit : Quelque soit E > 0 il existe. y > 0 tel que, quelque soit 5 rW tel que\%-a1 3 7, on ait la propriété P : \ &a) - fia> 1 s E .

C'est le théorème 3 qui nous assure que cette relation est bien. antisymétrlque, c'est-à-dire est bien une relation d'ordre; Il nous assure en outre que c'est une relation d'ordre totale : deux cardinaux sont nécessairement comparables. - Notons que, si E etr‘ sont deux ensembles, et s'il existe une surjection 4 de E sur F F est moins puissant En effet l'image rec;proque de tout point deT que E est no; vide, et si, dans chacune de ces images réclproques, nous choisissons exactement un élément, nous formons une partie de E qui est équlpotente à P * .

Un ensemble s'appelle aussi parfois un espace, ses él6ments s'appellent souvent des points. 4 O n p T n ol a ( d t 'p r E d eEe e. a ) e s nr C u n ' e n o q e np fu u s as e p ov d' Et . E m ) e c d - S % e f c xc p e d é Y s id tp o Xde E ,a o n d e qu x ren t i ux ts ï ots t e és nei o Y o o q a Y cn u u Xn e d sY , le o ée t X étY n cY 3 X C m . ' A r e oi #i X z * L d n a A s C. Si C XcY EcE 'é c de Ee l q od o r [ is n sa rA= e s'écrit C A ou s a c O ar r, * A S A e 6 s id ti o p a Aa - 6 p u d d A q n e u ' p O p à l ; * é a d A c n ' e A , é, C p d 5 1 d d A P ?

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Cours d'analyse : Vol. 1 by Laurent Schwartz

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