Download PDF by Allan M. Krall (auth.): Applied Analysis

By Allan M. Krall (auth.)

ISBN-10: 9027723427

ISBN-13: 9789027723420

ISBN-10: 9400947488

ISBN-13: 9789400947481

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Sample text

We have already encountered norms briefly in Chapter I. They are, roughly speaking, functions which measure distances between elements of a linear space. They must, of course, have certain characteristics. It is these which we now make precise. 1. DEFINITION. Let X be a linear space. Let x and y be in X. and let ~ be a complex (real) number. m on X. denoted by II-II. is a function whose domain is X. whose range is the nonnegative real numbers. and which satisfies 1. IIx+YII S IIxll + IIYU. 2. lI~xll = I~I 3.

I I / .... / I I " I I I I I I I I I I I I I I I I Isoclines: / t I // / ,. :; ....... " , / . / / / / I 11111/ I I I I I I I I I I I I I I I I I I I I I I I / I = x2 , CHAPTER III 50 We next show that solutions to IVP do exist with some regularity. The theorem we present is associated with the French mathematician Picard. 3. Let THEOREM (PICARD'S EXISTENCE THEOREM). ) be a real valued function which is continuous on D, and let f satisfy a Lipschitz condition in the second variable, Then there exists an interval I with T in such that IVP: x' = f(t,x), X(T) = (C 1 ) has a unique.

9. and. since A is Yn = xn/nllxnll X' we see that becomes larger. Yn aproaches continuous. A9 = 9' . O. Then for But 11 9 'II y = O. Since and we have a contradiction. A is bounded. then x in X. We replace x by x-y to find A satisfies a Lipschitz condition with Lipschitz constant IIAII. Finally. if A satisfies a Lipschitz condition. it is obviously continuous. The operator norm has a number of interesting properties. We state some of them for linear operators which transform the space X into itself.

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Applied Analysis by Allan M. Krall (auth.)

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