By Sze-Tsen Hu

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This implies, by the definition of the norm in the product of spaces, that inf suplltzll ~ z;eo teR zeXxN inf suplltPzll = +oo. z;e 0 teR zeXxN Thus X 0 is a {JF*-space. Observe that (ym, 0) = lim Zm, n· Hence (ym,O) E X 0 • Therefore (O,xm,n) is dense in Xx / 2 • = n(zm,n-(ym,O)) E X 0 • Thus X 0 D II ID be a finite-dimensional F-space. , let lltxll be non-decreasing for t > 0 and each x o:j::. 0. 6 (Kalton, 1974). Let (X, c = inf suplltxll· xex teR x;eo 53 Linear Operators Then,Jor all a < c, the set K = {x: llxll ~ a} is compact.

Chapter 1 30 Let t 0 e Q and let e be an arbitrary positive number. Let m be a positive integer such that 112m < e/4. Since the functions Xn(t) are continuous, there is a neighbourhood V of the point t0 such that, for t e V, lxn(t)-xn(to) e I<2 (n = 1,2, ... 2,; 2~ lxn(t)-xn(t )1. J. Since Q is compact, there is a continuous function x(t) such that lx(t)l < 1 x(t0 ) = 0, x(t) = I forte V, fort¢ V. J), therefore there is Xn such that llxn-xll = suplxn(t)-x(t)l tEV This implies that, if lxn(t)l lxn(t)-xn(t0)1 1 < -4 .

We have thus extended the operator A from the domain DA onto its closureDA· This is the reason why in the theory of continuous linear operators we shall restrict ourselves to the operators defined on the whole space X. Let X be a metric linear space. A set B C X is said to be bounded if, for any sequence of scalars { tn} tending to 0 and for any sequence {Xn} of elements of B, the sequence {tnxn} tends to 0. In other words, a set Biscalled bounded if, for any neighbourhood of zero U, there is a number b such that B C bU.

### Elements of General Topology by Sze-Tsen Hu

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