By Ward Cheney

This booklet advanced from a direction at our college for starting graduate stu dents in mathematics-particularly scholars who meant to concentrate on ap plied arithmetic. The content material of the direction made it appealing to different math ematics scholars and to graduate scholars from different disciplines reminiscent of en gineering, physics, and laptop technological know-how. because the direction used to be designed for 2 semesters period, many issues can be incorporated and handled in de tail. Chapters 1 via 6 replicate approximately the particular nature of the path, because it used to be taught over a few years. The content material of the direction used to be dictated by way of a syllabus governing our initial Ph. D. examinations within the topic of ap plied arithmetic. That syllabus, in flip, expressed a consensus of the college participants inquisitive about the utilized arithmetic application inside of 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 publication includes extra chapters having vital fabric now not integrated within the path: bankruptcy eight, on degree and integration, is for the ben efit of readers who desire a concise presentation of that topic, and bankruptcy 7 comprises a few issues heavily allied, yet peripheral, to the crucial thrust of the path. This association of the cloth merits a few explanation.

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15. - 1 7. Let A and be nonempty closed sets in a normed linear space. Prove that if A + B is compact , then so are A and B. Why do we assume that the sets are nonempty? Prove that if A is compact, then A + B is closed. B Norm ed Lin ear Cl1apter 1 24 Spaces 1 . 5 L inear Transformat ions Consider two vector spaces X and Y over the same scalar field. A mappi ng f : X � Y is said to be linear if f(o:u+ {3v) == o:f (u) + /3/(v) for all scalars o: and /3 and for al l vectors u , v in X. A li near map is often called a li near t ransformat ion or a linear operator .

Prove that the norm of a linear transformation is the infimum of all the numbers satisfy the inequality II Tx l l � M ll x ll for all x. 1\1 that 9. Prove the {surprising) result that a linear transformation is continuous if and only if it transforms every sequence converging to zero into a bounded sequence. 10. If f is a linear functional on X and N is its kernel, then there exists a one-dimensional subspace Y such that X = Y E9 N . (For two sets in a linear space, we define U + V as the set of all sums u + v when u ranges over U and v ranges over V.

28 . �� x(n}u(n} : u E l1 , ll ull 1 � 1 } 29. Use the Axiom of Choice to prove that for any set S having at least 2 points there is a function f : S --+ S that does not have a fixed point . 30. An interesting Banach space is the space c consisting of all convergent sequences. The norm is llxlloc == supn l x ( n ) l . Obviously, we have these set inclusions among the examples encountered so far: l1 C co C c C lx Prove that co is a hyperplane in c. Identify in concrete terms the conjugate space c• .