The strict topology and spaces with mixed topologies
HTML articles powered by AMS MathViewer
- by J. B. Cooper PDF
- Proc. Amer. Math. Soc. 30 (1971), 583-592 Request permission
Abstract:
In a paper about twenty years ago, and later papers, R. C. Buck introduced a new topology, the strict topology, on spaces of continuous functions on locally compact spaces. Since then, a considerable amount of work has been done on these and similar topologies by, among others, Conway, Collins, Rubel and Shields (see references [2], [3], [4], [6], [7], [15]). In the early nineteen-fifties, the Polish mathematicians, Alexiewicz and Semadeni, considered a vector space E, on which two norms are defined, and defined a notion of convergence of sequences on E which in some sense mixed the topologies given by the norms ([1] and later papers). In 1957, Wiweger [17] showed that under natural restrictions, the space E could be given a locally convex space structure where convergent sequences were precisely the sequences considered by Alexiewicz and Semadeni. Since then, this method of mixing topologies has been studied and generalised by several mathematicians ([18], [9]). The purpose of this note is to show that the strict topology for function spaces is a special case of a mixed topology. We then intend to use the theory of mixed topologies to give quick proofs of the basic results on strict topologies. This has the advantage of giving simpler proofs and of eliminating some heavy analysis. It also allows the definition of strict topologies in a more general setting than has been considered until now and in some cases gives new results for the standard situation.References
- A. Alexiewicz, On the two-norm convergence, Studia Math. 14 (1953), 49β56 (1954). MR 61755, DOI 10.4064/sm-14-1-49-56
- R. Creighton Buck, Operator algebras and dual spaces, Proc. Amer. Math. Soc. 3 (1952), 681β687. MR 50180, DOI 10.1090/S0002-9939-1952-0050180-5
- R. Creighton Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95β104. MR 105611 β, Algebraic properties of classes of analytic functions, Seminar on Analytic Functions, vol. II, Princeton University, Princeton, N. J., 1957, pp. 175-188.
- Heron S. Collins, On the space $l^{\infty }\,(S)$, with the strict topology, Math. Z. 106 (1968), 361β373. MR 239406, DOI 10.1007/BF01115085
- H. S. Collins and J. R. Dorroh, Remarks on certain function spaces, Math. Ann. 176 (1968), 157β168. MR 222644, DOI 10.1007/BF02056983
- John B. Conway, The strict topology and compactness in the space of measures. II, Trans. Amer. Math. Soc. 126 (1967), 474β486. MR 206685, DOI 10.1090/S0002-9947-1967-0206685-2
- J. B. Cooper, On a generalisation of Silva spaces, Math. Ann. 182 (1969), 309β313. MR 253010, DOI 10.1007/BF01350706
- D. J. H. Garling, A generalized form of inductive-limit topology for vector spaces, Proc. London Math. Soc. (3) 14 (1964), 1β28. MR 161121, DOI 10.1112/plms/s3-14.1.1
- Alexandre Grothendieck, Sur les espaces ($F$) et ($DF$), Summa Brasil. Math. 3 (1954), 57β123 (French). MR 75542 β, Espaces vectoriels topologiques, Inst. Mat. Pura Appl., Univ. SΓ£o Paulo, 1954. MR 17, 1110.
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- D. A. RaΔkov, A criterion of completeness of locally convex spaces, Uspehi Mat. Nauk 14 (1959), no.Β 1 (85), 223β229 (Russian). MR 0105000
- L. A. Rubel and J. V. Ryff, The bounded weak-star topology and the bounded analytic functions, J. Functional Analysis 5 (1970), 167β183. MR 0254580, DOI 10.1016/0022-1236(70)90023-6
- L. A. Rubel and A. L. Shields, The space of bounded analytic functions on a region, Ann. Inst. Fourier (Grenoble) 16 (1966), no.Β fasc. 1, 235β277. MR 198281
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- A. Wiweger, A topologisation of Saks spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 773β777, LXVII (English, with Russian summary). MR 0088689
- A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47β68. MR 133664, DOI 10.4064/sm-20-1-47-68
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 583-592
- MSC: Primary 46.01
- DOI: https://doi.org/10.1090/S0002-9939-1971-0284789-2
- MathSciNet review: 0284789