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By Ward Cheney

This ebook advanced from a direction at our collage for starting graduate stu­ dents in mathematics-particularly scholars who meant to specialise in ap­ plied arithmetic. The content material of the path made it beautiful to different math­ ematics scholars and to graduate scholars from different disciplines corresponding to en­ gineering, physics, and computing device technology. because the direction was once designed for 2 semesters period, many issues may be incorporated and handled in de­ tail. Chapters 1 via 6 mirror approximately the particular nature of the direction, because it was once taught over a couple of years. The content material of the path was once dictated via a syllabus governing our initial Ph. D. examinations within the topic of ap­ plied arithmetic. That syllabus, in flip, expressed a consensus of the school contributors considering the utilized arithmetic software inside our division. The textual content in its current manifestation is my interpretation of that syllabus: my colleagues are innocent for no matter what flaws are current and for any inadvertent deviations from the syllabus. The ebook includes extra chapters having very important fabric now not integrated within the direction: bankruptcy eight, on degree and integration, is for the ben­ efit of readers who desire a concise presentation of that topic, and bankruptcy 7 includes a few themes heavily allied, yet peripheral, to the imperative thrust of the path. This association of the fabric merits a few explanation.

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Thus one can have E C X C Z, where E is of category II in X and of category I in Z. For a concrete example, the one in the preceding paragraph will serve. The Corollary implies that if X is a complete metric space, then X is of the second category in X . " ( See Problems 5, 6, 7, for example. ) Theorem 2. Let { A0 } be a family of continuous linear transformations defined on a Banach space X and taking values in a normed linear space. In order that sup0 IAa ll < oo, it is necessary and sufficient that the set { x E X : sup0 I A a x ll < oo} be of the second category in X .

Since F = X, the precedi ng corollary implies that F is of the second category in X . For the sufficiency, define Fn = { x E X : sup 0 II Ao x ll � n} 42 Chapter 1 Normed Linear Spaces and assume that F is of the second category in X . Notice that F = U � Fn · Since F is of the second category, and each Fn is a closed subset of X ,1 the defini tion of second category implies that some Fm contains a bal l. Suppose that ( r > 0) B - {x E X : ll x - xo ll � r } C Fm For any x satisfying l l x ll IIAa x ll Hence I I Aa ll � 1 we have Hence == II Ao [r - 1 (xo + rx - xo)) I I � r - 1 IIA0 { xo + r x ) l!

On the space C(O, 1) we define "point-evaluation functionals" by t• (x) = x( t). Here t E [0, 1] and x E C(O, 1]. Prove that l it* I I = 1 . \i t; , where tt , t2 , . . \ i l · 20. In the proof of the Neumann Theorem we used the inequality II A k l l � II A I l k . Prove this. 2 1 . Prove that if { ¢ 1 , . . , ¢n } is a linearly independent set of linear functionals, then for suitable x1 we have cPi (Xj ) = � ij for 1 � i , j � n . 30 Chapter 1 Normed Linear Spaces 22. Prove that if a linear transformation is discontinuous at one point, then it is discontinuous everywhere.

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