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By Young N.

ISBN-10: 0521330718

ISBN-13: 9780521330718

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1 INTRODUCTION In discussing any branch of mathematics it is helpful to use the notation and terminology of set theory. This subject, which was developed by Boole and Cantor in the latter part of the 19th century, has had a profound influence on the development of mathematics in the 20th century. It has unified many seemingly disconnected ideas and has helped reduce many mathematical concepts to their logical foundations in an elegant and systematic way. We shall not attempt a systematic treatment of the theory of sets but shall confine ourselves to a discussion of some of the more basic concepts.

38. 1(0, 0)1 = Q, and IxI > 0 if x ; 0. iii) 1x/y1 = Ixl/IYI, if y # 0. i) ii) Ixyl = IxI IYI. iv) I(xl, 0)1 = lxii. Proof Statements (i) and (iv) are immediate. To prove (ii), we write x = x1 + ix2, y = y1 + iy2, so that xy = x1 y1 - x2y2 + i(x1 y2 + x2 y1). Statement (ii) follows from the relation Ixy12 = xiyi + xiy2 + xiy2 + x2yi = (x1 + x2)(y1 + y2) = IxI21Y12 Equation (iii) can be derived from (ii) by writing it in the form IxI = IYI Ix/yl Geometrically, IxI represents the length of the segment joining the origin to the point x.

15. A set S is called countable if it is either finite or countably infinite. A set which is not countable is called uncountable. The words denumerable and nondenumerable are sometimes used in place of countable and uncountable. 16. Every subset of a countable set is countable. S. If A is finite, there is Proof. Let S be the given countable set and assume A nothing to prove, so we can assume that A is infinite (which means S is also infinite). Let s = be an infinite sequence of distinct terms such that S = {s,, S2, }.

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An Introduction to Hilbert Space by Young N.

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