Abstract
This chapter starts recalling the definitions of Hausdorff topological space, metric space, and normed space. Examples of Banach sequence spaces, of continuous functions and of measurable functions, are given. Seminorms are introduced and they are used to introduce the concept of Hausdorff locally convex space. Bounded sets are defined and their properties are explained. The continuity of linear operators and the equicontinuity of sets of operators between locally convex spaces are characterized. The topological dual of a locally convex space is defined. Special attention is given to complete metrizable locally convex spaces, called Fréchet spaces. Spaces of continuous functions on open subsets of \(\mathbb {R}^N\) and Köthe echelon spaces are presented as examples of Fréchet spaces. Bounded subsets of Köthe echelon spaces are characterized. The Kolmogorov theorem concerning normable locally convex spaces is presented. This chapter concludes with the proof of two important theorems about spaces of continuous functions: Stone–Weierstraß theorem and Ascoli theorem. An appendix presents the main points about Hilbert spaces, including proofs of Riesz representation theorem, the Riesz–Fischer theorem, and the completeness of the exponential orthonormal system in the space of square integrable functions.
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References
Bachman, G., Narici, L.: Functional Analysis. Dover Publications, Inc., Mineola, NY (2000). Reprint of the 1966 original
Banach, S.: Théorie des Opérations Linéaires. Éditions Jacques Gabay, Sceaux (1993). Reprint of the 1932 original
Bogachev, V.I., Smolyanov, O.G.: Topological Vector Spaces and Their Applications. Springer Monographs in Mathematics. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57117-1
Brezis, H.: Functional analysis. In: Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)
Cerdà, J.: Linear functional analysis. In: Graduate Studies in Mathematics, vol. 116. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid (2010). https://doi.org/10.1090/gsm/116
Conway, J.B.: A course in functional analysis. In: Graduate Texts in Mathematics, vol. 96. Springer, New York (1985). https://doi.org/10.1007/978-1-4757-3828-5
Diestel, J.: Sequences and series in Banach spaces. In: Graduate Texts in Mathematics, vol. 92. Springer, New York (1984). https://doi.org/10.1007/978-1-4612-5200-9
Edwards, R.E.: Functional Analysis. Dover Publications, Inc., New York (1995). Theory and applications, Corrected reprint of the 1965 original
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). https://doi.org/10.1007/978-1-4419-7515-7
Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J., Zizler, V.: Functional analysis and infinite-dimensional geometry. In: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8. Springer, New York (2001). https://doi.org/10.1007/978-1-4757-3480-5
Floret, K., Wloka, J.: Einführung in die Theorie der lokalkonvexen Räume. Lecture Notes in Mathematics, No. 56. Springer, Berlin/New York (1968)
Grothendieck, A.: Topological Vector Spaces. In: Notes on Mathematics and Its Applications. Gordon and Breach Science Publishers, New York/London/Paris (1973). Translated from the French by Orlando Chaljub
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition
Heuser, H.G.: Functional Analysis. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester (1982). Translated from the German by John Horváth
Horváth, J.: Topological Vector Spaces and Distributions, vol. I. Addison-Wesley Publishing Co., Reading, Mass./London-Don Mills, Ont. (1966)
Jameson, G.J.O.: Topology and Normed Spaces. Chapman and Hall, London; Halsted Press, New York (1974)
Jarchow, H.: Locally Convex Spaces. Mathematische Leitfäden. B. G. Teubner, Stuttgart (1981)
Kadets, V.: A Course in Functional Analysis and Measure Theory. Universitext. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-92004-7. Translated from the 2006 Russian edition
Kelley, J.L.: General topology. In: Graduate Texts in Mathematics, No. 27. Springer, New York/Berlin (1975). Reprint of the 1955 edition
Köthe, G.: Topological vector spaces. I. Translated from the German by D. J. H. Garling. In: Die Grundlehren der Mathematischen Wissenschaften, Band 159. Springer, New York Inc., New York (1969)
Köthe, G.: Topological vector spaces. II. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 237. Springer, New York/Berlin (1979)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. I. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92. Springer, Berlin/New York (1977). Sequence spaces
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. II. In: Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97. Springer, Berlin/New York (1979). Function spaces
Meise, R., Vogt, D.: Introduction to functional analysis. In: Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press, Oxford University Press, New York (1997). Translated from the German by M. S. Ramanujan and revised by the authors
Pérez Carreras, P., Bonet, J.: Barrelled locally convex spaces. In: North-Holland Mathematics Studies, vol. 131. North-Holland Publishing Co., Amsterdam (1987). Notas de Matemática, 113
Ransford, T.J.: A short elementary proof of the Bishop-Stone-Weierstrass theorem. Math. Proc. Camb. Philos. Soc. 96(2), 309–311 (1984). https://doi.org/10.1017/S0305004100062204
Robertson, A.P., Robertson, W.: Topological vector spaces. In: Cambridge Tracts in Mathematics, vol. 53, 2nd edn. Cambridge University Press, Cambridge/New York (1980)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)
Rudin, W.: Functional Analysis, 2nd edn. In: International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York (1991)
Schaefer, H.H., Wolff, M.P.: Topological vector spaces. In: Graduate Texts in Mathematics, vol. 3, 2nd edn. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-1468-7
Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Robert E. Krieger Publishing Co., Inc., Melbourne, FL (1986)
Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Dover Publications, Inc., Mineola, NY (2006). Unabridged republication of the 1967 original
Valdivia, M.: Topics in locally convex spaces. In: Notas de Matemática [Mathematical Notes], vol. 85. North-Holland Publishing Co., Amsterdam/New York (1982)
Wilansky, A.: Modern Methods in Topological Vector Spaces. McGraw-Hill International Book Co., New York (1978)
Wilansky, A.: Topology for Analysis. Robert E. Krieger Publishing Co., Inc., Melbourne, FL (1983). Reprint of the 1970 edition
Yosida, K.: Functional analysis. In: Classics in Mathematics. Springer, Berlin (1995). https://doi.org/10.1007/978-3-642-61859-8. Reprint of the sixth (1980) edition
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Bonet, J., Jornet, D., Sevilla-Peris, P. (2023). Locally Convex Spaces. In: Function Spaces and Operators between them. RSME Springer Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-031-41602-6_2
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