An introduction to functional analysis by Charles Swartz

By Charles Swartz

In line with an introductory, graduate-level path given by means of Swartz at New Mexico country U., this textbook, written for college students with a average wisdom of element set topology and integration thought, explains the rules and theories of useful research and their purposes, exhibiting the interpla

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An introduction to functional analysis

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Let 8 > 0. Set 5 = &(M. If Ilx - yll < S, IIT(x - y)II = IITx - Tyll s Mllx - yII < e and (i) holds. Definition 6. Let X, Y be TVS over F. L(X, Y) denotes the space of all linear, continuous mappings (called operators from X into Y. L(X, Y) is a vector space under the pointwise operations of addition and scalar multiplication. If X = Y, we often write L(X) = L(X, X). Linear Operators and Linear Functionals 46 If X and Y are semi-NLS, we define a semi-norm on L(X, Y) by IITII = sup{IITxII : IIxII <_ 1); this semi-norm is called the operator norm or uniform norm on L(X, Y).

For 1 < p < -, the dual of LL(S, E, µ) is linearly isometric to Lq(S, s, µ), where F E (Lv)' and f e Lq The correspondence espondence between 1/p + 1/q = 1. 7). 7). In particular, if u is counting measure on IN, we have Example 14. For 1 <_ p < co, the dual of l p is l q, where 1/p + 1/q = 1, 1 < p, and if p = 1, q = oo. The correspondence between y = (yk} E l q 00 and the corresponding linear functional fy e (I P)' is y, x> _ ykxk, k=1 P II=IIYIiq Example 15. 31). For this we require the integration of bounded E-measurable functions with respect to bounded finitely additive measures belonging to n ba(E).

Let the sequence with a If coordinates. (ti } 1 in the jth coordinate and 0 J be in the other is a scalar sequence which converges to 0 with n o 0, then the sequence sn = Jee is Cauchy but does not converge to j=1 an element of c00. The sequence (tj ) also furnishes an example of a sequence which converges to 0 but which is not X convergent. Example 16. Let 1 5 p < Co, lp consists of all scalar sequences (tj } CO satisfying II {t } IIp = ( I t. I P)1/P < 00. Ip is a B-space under the norm j=1 22 II Quasi-normed and Normed Linear Spaces lip (this is a special case of Example 22).

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