Representations for transformations continuous in the norm

Authors:
J. R. Edwards and S. G. Wayment

Journal:
Trans. Amer. Math. Soc. **154** (1971), 251-265

MSC:
Primary 28.50; Secondary 46.00

DOI:
https://doi.org/10.1090/S0002-9947-1971-0274704-4

MathSciNet review:
0274704

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Abstract: Riemann and Lebesgue-type integrations can be employed to represent operators on normed function spaces whose norms are not stronger than sup-norm by where is determined by the action of *T* on the simple functions. The real-valued absolutely continuous functions on [0, 1] are not in the closure of the simple functions in the *BV* norm, and hence such an integral representation of an operator is not obtainable. In this paper the authors develop a *v*-integral whose structure depends on fundamental functions different than simple functions. This integral is as computable as the Riemann integral. By using these fundamental functions, the authors are able to obtain a direct, analytic representation of the linear functionals on *AC* which are continuous in the *BV* norm in terms of the *v*-integral. Further, the *v*-integral gives a characterization of the dual of *AC* in terms of the space of fundamentally bounded set functions which are convex with respect to length. This space is isometrically isomorphically identified with the space of Lipschitz functions anchored at zero with the norm given by the Lipschitz constant, which in turn is isometrically isomorphic to . Hence a natural identification exists between the classical representation and the one given in this paper. The results are extended to the vector setting.

**[1]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[2]**R. J. Easton and D. H. Tucker,*A generalized lebesgue-type integral*, Math. Ann.**181**(1969), no. 4, 311–324. MR**1513279**, https://doi.org/10.1007/BF01350670**[3]**J. R. Edwards and S. G. Wayment,*A unifying representation theorem*, Math. Ann.**187**(1970), 317–328. MR**0270181**, https://doi.org/10.1007/BF01396462**[4]**J. R. Edwards and S. G. Wayment,*Integral representations for continuous linear operators in the setting of convex topological vector spaces*, Trans. Amer. Math. Soc.**157**(1971), 329–345. MR**0281867**, https://doi.org/10.1090/S0002-9947-1971-0281867-3**[5]**T. H. Hildebrandt,*Linear continuous functionals on the space (𝐵𝑉) with weak topologies*, Proc. Amer. Math. Soc.**17**(1966), 658–664. MR**0193490**, https://doi.org/10.1090/S0002-9939-1966-0193490-3**[6]**T. H. Hildebrandt,*Linear operations on functions of bounded variation*, Bull. Amer. Math. Soc.**44**(1938), no. 2, 75. MR**1563684**, https://doi.org/10.1090/S0002-9904-1938-06685-2**[7]**Ralph E. Lane,*The integral of a function with respect to a function*, Proc. Amer. Math. Soc.**5**(1954), 59–66. MR**0059346**, https://doi.org/10.1090/S0002-9939-1954-0059346-3**[8]**I. P. Natanson,*Teoriya funkciĭ veščestvennoĭ peremennoĭ*, Gosudarstv. Izdat. Tehn.-Teor. Lit.,], Moscow-Leningrad, 1950 (Russian). MR**0039790****[9]**E. J. Purcell,*Calculus with analytic geometry*, Appleton-Century-Crofts, New York, 1965.**[10]**F. Riesz,*Sur les opérations fonctionelles linéaires*, C. R. Acad. Sci. Paris**149**(1909), 974-977.**[11]**Don H. Tucker,*A note on the Riesz representation theorem*, Proc. Amer. Math. Soc.**14**(1963), 354–358. MR**0145334**, https://doi.org/10.1090/S0002-9939-1963-0145334-0**[12]**D. H. Tucker and S. G. Wayment,*Absolute continuity and the Radon-Nikodym theorem*, J. Reine Angew. Math.**244**(1970), 1–19. MR**0272978****[13]**David J. Uherka,*Generalized Stieltjes integrals and a strong representation theorem for continuous linear maps on a function space*, Math. Ann.**182**(1969), 60–66. MR**0247439**, https://doi.org/10.1007/BF01350164

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0274704-4

Keywords:
Generalized Stieltjes-type integral,
*v*-integral,
integral representation,
convex with respect to length,
fundamentally bounded,
Lipschitz function,
dual space,
vector-valued functions,
absolute continuity,
bounded variation,
semiabsolute continuity,
semibounded variation

Article copyright:
© Copyright 1971
American Mathematical Society