By Mary W Gray

ISBN-10: 020102568X

ISBN-13: 9780201025682

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**Sample text**

Putting xn = an -bn, Yn = an -cn in the last inequaltiy, we have {f lan-bnIPf'P + p(a, b)+p(a, c), 30 TRIO SPACES EXERCISES 1. E is the set of all ordered pairs x = (xu X2) of real numbers. Prove that p(X, y) = max {Ixl - YII, IX2 - Yall is a metric on E. 2. the set all functions x(t) continuous on the bounded closed interval a oE; t oE; b, Prove that p(x,y) = sup{lx(t)-y(t)l:tE:[a,b]} is a metric onE. 3. E is the set of all differentiable fWlCtions x(t) whose uerivatives are continuous on the bounded closed interval a oE; t oE; b, Prove that p(x,y) Ix(a) is a metric on E.

The frontier of A is the complement the open set lnt A u Ext A, and this, as we shall see later, implies that the frontier of A is a closed set. The boundary of a set A, denoted by BdA, is the part of the frontier of A belongs to ; hence A nA'-, = A'n Evidently, if A is closed, its frontier is its boundary. The boundary of a set contains no non-empty open set. For if BdA contained a non-empty open set 0, 0 would be contained in IntA is largest open of A, 0 ,'muId be contained Int A. impossible o s;; Bd A S;; ]'r A S;; (Int A)'.

A similar argument, with the same conclusion, can be applied { 00 }l/P when the metric is p(x, y) = ~ IXn - Ynl p , 1 wherep> l. 37. Some complete function spaces The space consists of all functions [a, b] with metric sup Ix(t)-y(t)l, te: [a,b] If {xn(t)} is a Cauchy there exists a positive mt,egl3r every that IXm(t)-xn(t)1 < e, whenever m > n ~ no, for all tE [a, b]. This is the Cauchy condition for uniform convergence. But a uniformly convergent sequence of continuous functions tends to a continuous limit.

### A radical approach to algebra by Mary W Gray

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