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Linear operations on summable functions
| Content Provider | Semantic Scholar |
|---|---|
| Author | Dunford, Nelson Pettis, B. J. |
| Copyright Year | 1940 |
| Abstract | In the last few years the work of Gelfand [17, 18], $ Kantorovitch [23, 24, 25], Dunford [9, 10], Vulich [24, 25, 42] and others has shown that in developing a representation theory for various classes of linear operations! among Banach spaces [1] effective use can be made of abstract functions and integrals, just as the general linear functional over certain 7i-spaces were earlier discovered to be representable in terms of numerical functions and the integrals of numerical functions. This is especially true for operations defined to a general 73-space X from a Lebesgue space, that is, from a space consisting of a class of Lebesgue-integrable numerical functions. To obtain representations for operations of this sort it was found that ready application could be made of various integrals of the Lebesgue type that have been defined for functions taking their values in X. In the present paper we wish to communicate a representation theory for several types of operators mapping a space L(S), consisting of the real functions that are Lebesgue-integrable over an abstract aggregate 5 with respect to a fixed class of subsets of 6" and a fixed measure function [29, 34], into an arbitrary 73-space X. The representations will be given in terms of abstract integrals and kernel integrals. The general approach is not new, for it is based on the methods introduced by Gelfand [18] and Dunford [9] to obtain such theorems when 5 is a bounded real interval. However, in order to extend these known results to the case of an arbitrary 5 new devices are required since the earlier results were proved by Euclidean methods. In most instances we have been able to make the extension; this has been accomplished by generalizing the Radon-Nikodym theorem [29, 34] to set functions taking their values in an adjoint space and by substituting for differentiation processes the use of convex sets. The class of operators recently introduced by Kakutani [22] and Yosida [44] under the name weakly completely continu- |
| Starting Page | 323 |
| Ending Page | 392 |
| Page Count | 70 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9947-1940-0002020-4 |
| Volume Number | 47 |
| Alternate Webpage(s) | http://www.ams.org/journals/tran/1940-047-03/S0002-9947-1940-0002020-4/S0002-9947-1940-0002020-4.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9947-1940-0002020-4 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
