Abstract
Continuity theory serves as infrastructure for more specialized mathematical theories such as ordinary differential equations, partial differential equations, integral equations, operator theory, dynamical systems, and global analysis. The infrastructure includes creation of new spaces out of given ones, extension theorems, existence theorems, inversion theorems, approximation theorems, factorization theorems, adjunctions (e.g., exponential laws), and (local) representation theorems. So it embodies a great variety of possible topics. The present book, while deliberately not encyclopedic, does include a systematic study of linear continuity—enough to provide a foundation for functional analysis (the linear part of continuity theory).
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Notes
- 1.
(giving a pane in the glass)
References
J.Adamek, H.Herrlich and G.E. Strecker, Abstract and concrete categories, Dover Publications, New York, 2004.
R.Beattie and H.-P.Butzmann, Convergence Structures and Applications to Functional Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 2002.
E. Binz, Continuous convergence on C(X), Lecture Notes in Mathematics 469, Springer-Verlag, Berlin-Heidelberg-New York, 1975.
E.Binz and H.H. Keller, Funktionenräume in der Kategorie der Limesräume, Ann. Acad. Sci. Fenn. Ser. A No. 383, 1–21, 1966.
N.Bourbaki, Topologie générale, Actualité Scientific et Industrielles 1045,1084 1142,1143,1196,1235, Hermann, Paris 1953–1961.
N.Bourbaki, Espaces vectoriels topologiques, Actualité Scientific et Industrielles 1189,1229,1230, Hermann, Paris 1953–1955.
H.-P. Butzmann, An incomplete function space, Applied Categorical Structures, 9, 365–368, 2001.
E. Dubuc, Kan extensions in enriched category theory, Lecture Notes in Mathematics 145, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
N. Dunford and J.T. Schwartz, Linear Operators, Part 1, Interscience Publishers, New York, 1967.
L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, 1960.
J. Horváth, Topological vector spaces and distributions Volume I, Addison-Wesley, Reading Ma, 1966.
E. Hewitt, On two problems of Urysohn, Ann. of Math. 47 503–509 1946.
S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York, 1971.
L.D. Nel, Enriched algebraic structures with applications in Functional Analysis, Categorical Aspects of Topology and Analysis, Lecture Notes in Math. (Springer) 915, 247–259, 1982.
L.D. Nel, Upgrading functional analytic categories, Proc. Toledo Int. Conf. 408–424, 1983.
L.D. Nel, Infinite Dimensional Calculus allowing nonconvex domains with empty interior, Mh. Math. 110, 145–166, 1990.
L.D. Nel, Differential Calculus founded on an Isomorphism, Applied Categorical Structures, 1, 51–57, 1993.
S. Willard, General Topology, Addison-Wesley, Reading, Ma, 1970.
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Nel, L. (2016). Overview. In: Continuity Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-31159-3_1
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DOI: https://doi.org/10.1007/978-3-319-31159-3_1
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