Download e-book for kindle: An Introduction to Abstract Analysis by W. A. Light (auth.)

By W. A. Light (auth.)

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4 we need to calculate the values of a and b. Then the largest possible value for 6 will be the smallest of 1/n- a and b -1/n. We have 1 1 1 --a= - - - n n n+1 while 1 b-n 1 1 1 = -2 , n +n 1 =--- = -. n- 1 n n2 - n Some simple results 21 Thus the largest possible value we may take for 6 is 1/(n2 + n). , n larger), we see that we require a smaller and smaller value for 6 in order to retain the inequality lf(x)- /(1/n)l < 1. More important is the fact that there is no single value of 6 which will work for all values of y.

If we encounter the notation f- 1 (3) for a function from 1R to 1R, then we would interpret this as implying that f is an invertible function and the notation as referring to the inverse image of 3 under the function f. For the same function the notation f- 1 ([0, 1]) is not to be interpreted as implying anything about invertibility of f. It simply stands for the set of real numbers which are mapped into [0, 1J by f. If we want to denote the set of real numbers which are mapped onto the point 3 by a (not necessarily invertible or one-to-one) function f then we would write 1 ( {3}) to distinguish it from f- 1 (3).

Then the following sets are open: 1. r) ,8}, {x: a< f(x) < ,8}. The following sets are closed {x: f(x)::; a}, {x: f(x) 2: ,8}, {x: a::; f(x)::; ,8}. What can you say about the set {x: a< f(x)::; ,8}? 2. Obtain short proofs based on 3. 7 for questions 8 and 9 in the previous set of exercises. 3. Show that {(s,t): s 2 + t 2 < 1} is open in lR 2 with any of the three norms II· ll1, II· ll2 or ll·lloo· 4. Show that in lR3 , (i) the set {(s,t,u): s,t,u > 0} is open (ii) the set {(s,t,u): s = t = u} is not open.

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