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

By W. A. Light (auth.)

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Set Z = (X x Y, II · liz), where as usual X x Y is the set of ordered pairs (x, y ), where :r lies in X and y in Y. The norm is defined by ll=llz = ll(x,y)liz = max{llxllx, IIYIIY }. For sets A E X and B E Y show that (i) A x B is open if and only if A and B are open (ii) A x B is closed if and only if A and B are closed. Open and closed sets 41 14. The following constructs the Cantor set in R: begin with the unit interval [0, 1]. The set F1 is obtained from this set by removing the middle third, so that Now F 2 is obtained from F 1 by removing the middle third from each of the constituent intervals of F 1 .

V) Show that {an} converges. 5. The following shows how to determine that the linear space C[O, 1] with the norm is incomplete. (a) Define x n { s) = -1 s E [-1, -~] { ns s E (- ~, ~) 1 sE[~,1] Sketch Xn and compute llxp- Xq II for arbitrary values of p and q. (b) Show that {Xn} is a Cauchy sequence. (c) Define x as the function x(s) ={ -1 s E [-1,0) 1 sE[O,l]. Compute llx- xnll for abitrary values of n in IN. (d) Show that llxn- xll --+ 0 as n--+ oo. (e) Show that C[O, 1] with the given norm is not complete.

Define h, k : 1Rn --+ 1R by 4. Let h(x) = sup{f(x),g(x)} and k(x) = inf{f(x),g(x)}, x E 1Rn. Show that h and k are continuous at a. 5. Let F: 1R2 --+ 1R be defined by F(s, t) = { s02 + t2 s, t E

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