Calculus

An Introduction to Abstract Analysis by W. A. Light (auth.) PDF

By W. A. Light (auth.)

ISBN-10: 0412310902

ISBN-13: 9780412310904

ISBN-10: 1489972544

ISBN-13: 9781489972545

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Example text

However, one consequence of some rather powerful theorems which come later in this book will be that all continuous functions are uniformly continuous on a wide class of subsets of a normed linear space. For example, we shall show that every continuous function from R to R is uniformly continuous on each closed interval in R. The previous discussion can now be used to see that the function f : R --+ R defined by f(x) = 1/x for x # 0 and /{0) = 0 is not uniformly continuous on the set A= {0, 1).

8. Let f be a continuous, real-valued function on R. Show that the set {(a, b): b ~ f(a)} is closed in R 2 . 9. Let f and g be continuous mappings from a normed linear space X into a. second normed linear space Y. Prove that {x : /( x) = g(x)} is closed in X. 36 Open and closed sets 10. Show that {(a,:) : a > 0} is closed in R 2 • Open and closed sets can be used to give an alternative definition of the continuity of a function. This new viewpoint is potentially very attractive, because it removes any mention of the quantifiers f and 8, which on first encounter prove to be somewhat awkward to work with.

24 Some simple results 9. Prove that the elements of a convergent sequence in a normed linear space always form a bounded set. 10. Let R have the usual norm. Calculate (i) dist(1,{t: t 2 > 2}) (ii) dist(O,{t: t = 1/n, n EN}). 11. Let X= R 2 with the norm defined by li(s, t)ll = lsi+ it!. Let A = {(s, t) : s = t}. What is dist(y, A), when y Describe the set of best approximations to y from A. = (0, 1)? 12. Let X= C[-1, 1] with norm llxll = sup{lx(t)l : t E [-1, 1]}, for x EX. Let A be the set of constant functions in C[-1, 1].

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An Introduction to Abstract Analysis by W. A. Light (auth.)


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