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Additional info for Analysis 2: Differentialrechnung im IRn, gewöhnliche Differentialgleichungen, 9. Auflage

Sample text

Wegen der gleichmäßigen Konvergenz existiert ein NE N, so dass II/N(X),j(X)1I< ~ für alle x EX. d. Lineare Abbildungen Satz 11. Seien V und W normierte Vektorräume (über JR oder C) und sei A:V----tW eine lineareAbbildung. A ist genau dann stetig, wenn es eine reelle Konstante C ~ 0 gibt, so dass IIA(x) 11 ~ CIIxll für alle xE V. § 2 Grenzwerte. Stetigkeit 25 Beweis. a) Wir setzen zunächst die Stetigkeit von A im Nullpunkt voraus. Dann gibt es zu E = I ein 0 > 0, so dass IIA(z) 11 < I fiir alle z E V mit Ilzll < o.

Dann gilt für die Kurve g = /o

Die Umkehrabbildung I-I: Y Punkt (1,0) E Y. Die Punktfolge Pk:= (cos(21t-1/k),sin(21t-1/k)) E Y, -> X ist aber unstetig im k~ 1, konvergiert für k -> 00 gegen den Punkt (1,0) = 1(0), die Folge r l (pk) = 21t- 1/ k, k~1, konvergiert aber nicht gegen 0. Man kann sogar zeigen (vgl. § 3), dass es überhaupt keinen Homöomorphismus X -> Y gibt. I. Differentialrechnung im JRn 24 Gleichmäßige Konvergenz von Funktionenfolgen Definition. Seien X eine beliebige Menge, Y ein metrischer Raum , sowie In : X ----t Y, nE N, und I: X ----t Y Abbildungen.

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