By Graeme L. Cohen

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

Cover; Half-title; Series-title; identify; Copyright; Contents; Preface; 1 Prelude to fashionable research; 2 Metric areas; three The fastened 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 area; Bibliography; chosen suggestions; Index.

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**Example text**

We choose to use sequences to generate much of our theory, for the reasons mentioned above, but there is a common alternative based on the more primitive notion of open sets. This approach usually begins with the concept of a topological space, which is quite an early notion in the hierarchy of spaces indicated above. We in effect will be simplifying things a little by starting some way along the hierarchy, though later, in Chapter 5, we will pull the various approaches together. Now back to work.

If it does include ~ T , then simply remove it from the set so that the remaining n —1 sets are the required sub covering of T. □ Now we come to the Heine-Borel theorem. This is like the BolzanoWeierstrass theorem in that it describes a fundamental property of the real number system— fundamental because it is very closely related to the axiomatic concept o f completeness. 6. 7 (H ein e—B o r el T h e o re m ) A point set is compact if and only if it is closed and bounded. 4, that compact subsets of R are closed and bounded, so here we must prove the converse, that subsets of R which are closed and bounded are compact.

If { an} has only a finite range, then the desired conclusion is easily obtained, and we omit the details. Otherwise, the point set {a i, a2, a%,. . 7 may be applied: the least upper bound must exist. Write £ = supan. For any e > 0, we must have ajv > £ — e for some N E N because £ — e cannot also be an upper bound for { a n}. But { an} is nondecreasing, so that ajv ^ &JV+ 1 ^ a N + 2 ^ •••? implying that an > £ — e for all n > N. Furthermore, an < £ < £ + e for all n, and in particular for all n > N.