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 techniques subject matters first and foremost via convergence of sequences in metric house. even though, the choice topological process can be defined. purposes are incorporated from differential and vital equations, platforms of linear algebraic equations, approximation conception, numerical research and quantum mechanics.

Cover; Half-title; Series-title; identify; Copyright; Contents; Preface; 1 Prelude to trendy research; 2 Metric areas; three The fastened element Theorem and its functions; 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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**Extra info for A course in modern analysis and its applications**

**Example text**

Now, Y X X n is the set of ordered pairs { ( ( x i , x 2 , . . , x n_ i ) , x n) : xjt G Xjt, k = 1, 2, . . , n }. The mapping f : Y x X n —>X\ x X 2 x •••x X n given by / ( ( ( x i , x 2, - -. , x n_ i ) , x n)) = ( x i , x 2, . , x n_ i , x n) IE = is clearly a one-to-one correspondence, and this establishes that X i X X 2 x ■ ■■ x X n, or is countable. The induction is complete, and (a) is proved. The proof of (b) uses a similar method of counting. As before, we write Xk = { xk i , xk2 ,xks, •••}, for k G N.

X n_ i , x n) IE = is clearly a one-to-one correspondence, and this establishes that X i X X 2 x ■ ■■ x X n, or is countable. The induction is complete, and (a) is proved. The proof of (b) uses a similar method of counting. As before, we write Xk = { xk i , xk2 ,xks, •••}, for k G N. 4 Countability 17 in the array X u * 1 2 —» * 1 3 1 / *21 / *22 / *14 / *23 *24 / *31 *32 *33 *34 1 / *41 *42 *43 *44 and count them in the order indicated (those whose subscripts total 2 , then 3, then 4, . . ) , this time taking care that any x ’s belonging to more than one Xk are counted only once.

It is therefore permissible, when speaking of a countable set X , to write X = { x i , x 2, x 3 ,. }, implying that any element of X will eventually be included in the list x\, x 2, £ 3, . . In proving below that a given set is countable, we will generally be satisfied to indicate how the set may be counted or listed, and will not give an actual mapping which confirms the equivalence of the set with N. For example, the set Z of all integers is countable, since we may write Z = {0, —1,1, —2,2, —3 , 3 , .

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