The strict topology and spaces with mixed topologies

Author:
J. B. Cooper

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

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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.

**[1]**A. Alexiewicz,*On the two-norm convergence*, Studia Math.**14**(1953), 49-56. MR**15**, 880. MR**0061755 (15:880a)****[2]**R. C. Buck,*Operator algebras and dual spaces*, Proc. Amer. Math. Soc.**3**(1952), 681-687. MR**14**, 290. MR**0050180 (14:290f)****[3]**-,*Bounded continuous functions on a locally compact space*, Michigan Math. J.**5**(1958), 95-104. MR**21**#4350. MR**0105611 (21:4350)****[4]**-,*Algebraic properties of classes of analytic functions*, Seminar on Analytic Functions, vol. II, Princeton University, Princeton, N. J., 1957, pp. 175-188.**[5]**H. S. Collins,*On the spaces*,*with the strict topology*, Math. Z.**106**(1968), 361-373. MR**39**#763. MR**0239406 (39:763)****[6]**H. S. Collins and J. R. Dorroh,*Remarks on certain function spaces*, Math. Ann.**176**(1968), 157-168. MR**36**#5694. MR**0222644 (36:5694)****[7]**J. B. Conway,*The strict topology and compactness in the space of measures*, Trans. Amer. Math. Soc.**126**(1967), 474-486. MR**0206685 (34:6503)****[8]**J. B. Cooper,*On a generalization of Silva spaces*, Math. Ann.**182**(1969), 309-313. MR**0253010 (40:6225)****[9]**D. J. H. Garling,*A generalized form of inductive-limit topology for vector spaces*, Proc. London Math. Soc. (3)**14**(1964), 1-28. MR**28**#4330. MR**0161121 (28:4330)****[10]**A. Grothendieck,*Sur les espaces*(*F*)*et*(*DF*), Summa Brasil. Math.**3**(1954), 57-123. MR**17**, 765. MR**0075542 (17:765b)****[11]**-,*Espaces vectoriels topologiques*, Inst. Mat. Pura Appl., Univ. São Paulo, 1954. MR**17**, 1110.**[12]**K. Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**24**#A2844. MR**0133008 (24:A2844)****[13]**D. A. Raĭkov,*A criterion of completeness in locally convex spaces*, Uspehi Mat. Nauk**14**(1959), no. 1 (85), 223-229. (Russian) MR**21**#3747. MR**0105000 (21:3747)****[14]**L. A. Rubel and J. V. Ryff,*The bounded weak star topology and the bounded analytic functions*, J. Functional Analysis**5**(1970), 167-183. MR**0254580 (40:7788)****[15]**L. A. Rubel and A. L. Shields,*The space of bounded analytic functions on a region*, Ann. Inst. Fourier (Grenoble)**16**(1966), fasc. 1, 235-277. MR**33**#6440. MR**0198281 (33:6440)****[16]**H. H. Schaefer,*Topological vector spaces*, Macmillan, New York, 1966. MR**33**#1689. MR**0193469 (33:1689)****[17]**A. Wiweger,*A topologisation of Saks spaces*, Bull. Acad. Polon. Sci. Cl. III.**5**(1957), 773-777. MR**19**, 564. MR**0088689 (19:564b)****[18]**-,*Linear spaces with mixed topology*, Studia Math.**20**(1961), 47-68. MR**24**#A3490. MR**0133664 (24:A3490)**

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0284789-2

Keywords:
Locally convex spaces,
two normed spaces,
mixed topologies,
strict topologies

Article copyright:
© Copyright 1971
American Mathematical Society