×

Bounded continuous functions on a completely regular space. (English) Zbl 0244.46027


MSC:

46A03 General theory of locally convex spaces
54D30 Compactness
54D50 \(k\)-spaces
54D60 Realcompactness and realcompactification
46A20 Duality theory for topological vector spaces
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46E05 Lattices of continuous, differentiable or analytic functions
46E25 Rings and algebras of continuous, differentiable or analytic functions
28A10 Real- or complex-valued set functions
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. D. Aleksandrov, Additive set-functions in abstract spaces, a) 8 (50) (1940), 307-348; b) ibid. 9 (51) (1941), 563-628; c) Mat. Sb. 13 (55) (1943), 169-238. MR 2, 315; MR 3, 207; MR 6, 275. · Zbl 0060.13502
[2] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 0271.60009
[3] R. Creighton Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95 – 104. · Zbl 0087.31502
[4] Lucien LeCam, Convergence in distribution of stochastic processes, Univ. Calif. Publ. Statist. 2 (1957), 207 – 236.
[5] John B. Conway, The strict topology and compactness in the space of measures, Bull. Amer. Math. Soc. 72 (1966), 75 – 78. · Zbl 0137.31503
[6] John B. Conway, The strict topology and compactness in the space of measures. II, Trans. Amer. Math. Soc. 126 (1967), 474 – 486. · Zbl 0166.10901
[7] J. B. Cooper, The strict topology and spaces with mixed topologies, Proc. Amer. Math. Soc. 30 (1971), 583 – 592. · Zbl 0225.46004
[8] J. R. Dorroh, The localization of the strict topology via bounded sets, Proc. Amer. Math. Soc. 20 (1969), 413 – 414. · Zbl 0176.43003
[9] R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc. 112 (1964), 483 – 507. · Zbl 0138.17401
[10] Xavier Fernique, Processus linéaires, processus généralisés, Ann. Inst. Fourier (Grenoble) 17 (1967), no. fasc. 1, 1 – 92 (French). · Zbl 0167.16702
[11] D. H. Fremlin, D. J. H. Garling and R. G. Haydon, On measures on topological spaces (to appear). · Zbl 0236.46025
[12] Robin Giles, A generalization of the strict topology, Trans. Amer. Math. Soc. 161 (1971), 467 – 474. · Zbl 0204.43603
[13] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. · Zbl 0093.30001
[14] E. Granirer, On Baire measures on \?-topological spaces, Fund. Math. 60 (1967), 1 – 22. · Zbl 0146.12204
[15] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. · Zbl 0040.16802
[16] J. Hoffmann-Jørgensen, A generalization of the strict topology and its applications to compactness in \( C(X)\) and \( M(X)\), Preprint Series 1969/70, no. 32, Matematisk Institut, Aarhus Universitet.
[17] M. Katětov, Measures in fully normal spaces, Fund. Math. 38 (1951), 73 – 84. · Zbl 0045.17101
[18] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. · Zbl 0066.16604
[19] R. B. Kirk, Measures in topological spaces and \?-compactness, Nederl. Akad. Wetensch. Proc. Ser. A 72=Indag. Math. 31 (1969), 172 – 183. · Zbl 0175.14702
[20] R. B. Kirk, Locally compact, \?-compact spaces, Nederl. Akad. Wetensch. Proc. Ser. A 72=Indag. Math. 31 (1969), 333 – 344. · Zbl 0188.28003
[21] J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc. (3) 17 (1967), 139 – 156. · Zbl 0154.05101 · doi:10.1112/plms/s3-17.1.139
[22] Jan Mařík, The Baire and Borel measure, Czechoslovak Math. J. 7(82) (1957), 248 – 253 (English, with Russian summary). · Zbl 0091.05501
[23] W. Moran, The additivity of measures on completely regular spaces, J. London Math. Soc. 43 (1968), 633 – 639. · Zbl 0159.07802 · doi:10.1112/jlms/s1-43.1.633
[24] -, Measures and mappings on topological spaces, Proc. London Math. Soc. (3) 19 (1969), 493-508. · Zbl 0186.37201
[25] W. Moran, Measures on metacompact spaces, Proc. London Math. Soc. (3) 20 (1970), 507 – 524. · Zbl 0199.37802 · doi:10.1112/plms/s3-20.3.507
[26] S. E. Mosiman and R. F. Wheeler, The strict topology in a completely regular setting: Relations to topological measure theory (submitted). · Zbl 0219.46003
[27] Claude Portenier, Théorème de Stone-Weierstrass et espaces de type (\?), C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A872 – A874 (French). · Zbl 0191.12702
[28] Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen. 1 (1956), 177 – 238 (Russian, with English summary). · Zbl 0075.29001
[29] D. A. Raĭkov, A criterion of completeness of locally convex spaces, Uspehi Mat. Nauk 14 (1959), no. 1 (85), 223 – 229 (Russian).
[30] A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. · Zbl 0123.30202
[31] A. C. M. van Rooij, Tight functionals and the strict topology, Kyungpook Math. J. 7 (1967), 41 – 43.
[32] Helmut H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966. · Zbl 0141.30503
[33] F. Dennis Sentilles, Conditions for equality of the Mackey and strict topologies, Bull. Amer. Math. Soc. 76 (1970), 107 – 112. · Zbl 0215.20201
[34] F. Dennis Sentilles, The strict topology on bounded sets, Pacific J. Math. 34 (1970), 529 – 540. · Zbl 0205.40901
[35] F. Dennis Sentilles and Donald Curtis Taylor, Factorization in Banach algebras and the general strict topology, Trans. Amer. Math. Soc. 142 (1969), 141 – 152. · Zbl 0185.21103
[36] Flemming Topsøe, Compactness in spaces of measures, Studia Math. 36 (1970), 195 – 222. (errata insert). · Zbl 0201.06202
[37] W. H. Summers, The general complex bounded case of the strict weighted approximation problem, Math. Ann. 192 (1971), 90 – 98. · Zbl 0203.43402 · doi:10.1007/BF02052753
[38] -, Separability in the strict and substrict topologies (submitted). · Zbl 0256.46037
[39] V. S. Varadarajan, Measures on topological spaces, Mat. Sb. (N.S.) 55 (97) (1961), 35 – 100 (Russian).
[40] J. H. Webb, Sequential convergence in locally convex spaces, Proc. Cambridge Philos. Soc. 64 (1968), 341 – 364. · Zbl 0157.20202
[41] R. F. Wheeler, Characterizations of the strict topology for completely regular spaces (submitted).
[42] A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47 – 68. · Zbl 0097.31301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.