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Unformatted text preview: Lecture 4 Definition 1. Let a be a point in the open subset U of the normed space X and let Y be a normed space. The function F : U → Y is said to be differentiable at a iff there exists a bounded linear map A ∈ L ( X,Y ) such that lim k h k→ F ( a + h ) F ( a ) A ( h ) k h k = 0 . More explicitly, this says that for each ε > 0, there exists δ > 0 such that k h k ≤ δ ⇒ k F ( a + h ) F ( a ) A ( h ) k ≤ ε k h k . We call A the derivative of F at a and denote it as F a . Exercise : If A exists, then it is unique. Proof. Let A 1 and A 2 be derivatives for F : U → Y at a . Then, given ε > k A 1 ( h ) A 2 ( h ) k = k ( F ( a + h ) F ( a ) A 2 ) ( F ( a + h ) F ( a ) A 1 ) k ≤ k F ( a + h ) F ( a ) A 2 k + k F ( a + h ) F ( a ) A 1 k ≤ 2 hε provided that k h k ≤ δ for some δ > 0. This implies that k A 1 A 2 k = 0 so that A 1 = A 2 and the derivative of F at a is unique. Theorem 1. Differentiability implies continuity....
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.
 Summer '10
 Staff
 Algebra

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