Let fbe a continuous function on an open subset U of a Banach space Xinto a Banach space Y. Let a and'b be two distinct points of U such that the segment[a, b] is contained in U, and fbe differentiable at all points of [a, b]. Then|| ƒ (b) - ƒ (a) || ≤ || b-a || Sup { || Df (x) || : x = [a, b]}.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 27E: 27. Let , where and are nonempty. Prove that has the property that for every subset of if and...

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Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
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Let fbe a continuous function on an open subset U of a Banach spe into a Banach space Y. Let a and b be two distinct points of U such that the seg [a, b] is contained in U, and fbe differentiable at all points of [a, b]. Then