Abstract
Bundles are traditionally employed for studying various algebraic systems in mathematical analysis. The technique of bundles is used in examining Banach spaces, Riesz spaces, C*-algebras, Banach modules, etc. (see, for instance, [3, 6, 7, 13–15]). Representation of some objects of functional analysis as spaces of sections of corresponding bundles serves as a basis for some theories valuable in their own right. One of these theories in [8–12] is devoted to the notion of a continuous Banach bundle (CBB) and its applications to lattice normed spaces (LNSs). Within this theory, in particular, a representation is obtained for an arbitrary LNS as a space of sections of a suitable CBB.
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References
Aliprantis C.D. and Burkinshaw O., Positive Operators, Academic Press, New York (1985).
Arkhangel’skiĭ A. V. and Ponomarëv V. I., Fundamentals of General Topology: Problems and Exercises, Reidel, Dordrecht etc. (1984).
Burden C. W. and Mulvey C. J., “Banach spaces in categories of sheaves,”in: Applications of Sheaves, Springer-Verlag, Berlin, 1979, pp. 169–196 (Lecture Notes in Math.; 753).
Diestel J., Sequences and Series in Banach Spaces,Springer-Verlag, Berlin etc. (1984) (Graduate Texts in Mathematics, 92).
Engelking R., General Topology, Springer-Verlag, Berlin etc. (1985).
Fourman M. P., Mulvey C. J., and Scott D. S. (eds.), Applications of Sheaves, Springer-Verlag, Berlin (1979) (Lecture Notes in Math.; 753).
Gierz G., Bundles of Topological Vector Spaces and Their Duality, Springer-Verlag, Berlin (1982) (Lecture Notes in Math.; 955).
Gutman A. E., “Banach bundles in the theory of lattice normed spaces. I. Continuous Banach bundles,” Siberian Adv. Math., 3, No. 3, 1–55 (1993).
Gutman A. E., “Banach bundles in the theory of lattice normed spaces. II. Measurable Banach bundles,” Siberian Adv. Math., 3, No. 4, 8–40 (1993).
Gutman A. E., “Banach bundles in the theory of lattice normed spaces. III. Approximating sets and bounded operators,” Siberian Adv. Math., 4, No. 2, 54–75 (1994).
Gutman A. E., “Locally one-dimensional K-spaces and a-distributive Boolean algebras,” Siberian Adv. Math., 5, No. 2, 99–121 (1995).
Gutman A. E., “Banach bundles in the theory of lattice normed spaces,” in: Linear Operators Coordinated with Order [in Russian], Trudy Inst. Mat. (Novosibirsk), Novosibirsk, 1995, 29, pp. 63–211.
Hofmann K. H., “Representations of algebras by continuous sections,” Bull. Amer. Math. Soc., 78, 291–373 (1972).
Hofmann K. H. and Keimel K., “Sheaf theoretical concepts in analysis: bundles and sheaves of Banach spaces, Banach C(X)-modules,” in: Applications of Sheaves, Springer-Verlag, Berlin, 1979, pp. 414–441.
Kitchen J. W. and Robbins D. A., Gelfand Representation of Banach Modules, Polish Scientific Publishers, Warszawa (1982) (Dissertationes Mathematicae; 203).
Koptev A. V., “A reflexivity criterion for stalks of a Banach bundle,” Siberian Math. J., 36, No. 4, 735–739 (1995) (Translated from Russian: Sibirsk. Mat. Zh. 1995. 36, No. 4, 851–857).
Kusraev A. G., Vector Duality and Its Applications [in Russian], Nauka, Novosibirsk (1985).
Kutateladze S. S., Fundamentals of Functional Analysis, Kluwer Academic Publishers, Dordrecht (1996).
Schochetman I. E., “Kernels and integral operators for continuous sums of Ba-nach spaces,” Mem. Amer. Math. Soc., 14, No. 202, 1–120 (1978).
Vulikh B. Z., Introduction to the Theory of Partially Ordered Spaces, Wolters-Noordhoff, Groningen, Netherlands (1967).
Wnuk W., Banach Lattices with Properties of the Schur Type. A Survey, Dipartimento di Matematica dell’Università di Bari, Bari (1993).
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Gutman, A.E., Koptev, A.V. (2000). Dual Banach Bundles. In: Kutateladze, S.S. (eds) Nonstandard Analysis and Vector Lattices. Mathematics and Its Applications, vol 525. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4305-9_3
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DOI: https://doi.org/10.1007/978-94-011-4305-9_3
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