By Graeme L. Cohen

ISBN-10: 0511061668

ISBN-13: 9780511061660

ISBN-10: 0511070128

ISBN-13: 9780511070129

Designed for one-semester classes for senior undergraduates, this e-book methods themes first and foremost via convergence of sequences in metric house. besides the fact that, the choice topological strategy can also be defined. functions are incorporated from differential and imperative equations, structures of linear algebraic equations, approximation concept, numerical research and quantum mechanics.

Cover; Half-title; Series-title; identify; Copyright; Contents; Preface; 1 Prelude to trendy research; 2 Metric areas; three The mounted element Theorem and its purposes; four Compactness; five Topological areas; 6 Normed Vector areas; 7 Mappings on Normed areas; eight internal Product areas; nine Hilbert house; Bibliography; chosen recommendations; Index.

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Since £ is the least cluster point for S, only finitely many points of S are less than or equal to £ —<5, and similarly only finitely many are greater than or equal to £ + <5. So all but a finite number of the points of S are within (£ — <5, £ + <5). Then either £ — 8 is a lower bound for S or the set {x : x sC £ — <5, x G is not empty but is finite. Either way, S is bounded below. Similarly, S is bounded above. We have proved the following theorem. 9 If a point set has a limit point, then it is bounded.

13 A complex-valued sequence { zn} is said to be convergent to £ if for any number e > 0 there exists a positive inte­ ger N such that \zn — C\ < e whenever n > N. We then write lim zn = £ or zn —> £ and call ( the limit of { zn}. Of course, £ may be a complex number. The rider ‘n —» oo’ is often added for clarification. There is no need to say more at this stage specifically about complex­ valued sequences. The point has been made that we are not able to set up a definition of convergence which exactly parallels that for real­ valued sequences, but nonetheless it is the real-valued theory which sub­ sequently suggests an adequate definition.

Does g - 1 exist? If so, write out the function in full. Does f o g exist? Does g o f exist? If so, write out the function in full. (2) Define a function / : R —>•R by f ( x ) = 5x — 2, for x G R. Show that / is one-to-one and onto. Find / - 1 . (3) For functions / : X —> Y and g : Y Z, show that (a) g o / : X —> Z is one-to-one if / and g are both one-to-one, (b) g o / : X —> Z is onto if / and g are both onto. 4 C ou n ta b ility Our aim is to make a basic distinction between finite and infinite sets and then to show how infinite sets can be distinguished into two types, called countable and uncountable.

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