Varieties of linear topological spaces
HTML articles powered by AMS MathViewer
- by J. Diestel, Sidney A. Morris and Stephen A. Saxon
- Trans. Amer. Math. Soc. 172 (1972), 207-230
- DOI: https://doi.org/10.1090/S0002-9947-1972-0316992-2
- PDF | Request permission
Abstract:
This paper initiates the formal study of those classes of locally convex spaces which are closed under the taking of arbitrary subspaces, separated quotients, cartesian products and isomorphic images. Well-known examples include the class of all nuclear spaces and the class of all Schwartz spaces.References
- Richard F. Arens and James Eells Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397–403. MR 81458 S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN , Warsaw, 1932; reprint, Chelsea, New York, 1955. MR 17, 175.
- C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
- C. Bessaga and A. Pełczyński, A topological proof that every separable Banach space is homeomorphic to a countable product of lines, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 487–493 (English, with Russian summary). MR 257995
- C. Bessaga, A. Pełczyński, and S. Rolewicz, On diametral approximative dimension and linear homogeneity of $F$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 677–683. MR 132374
- M. S. Brooks, Sidney A. Morris, and Stephen A. Saxon, Generating varieties of topological groups, Proc. Edinburgh Math. Soc. (2) 18 (1972/73), 191–197. MR 318383, DOI 10.1017/S0013091500009913
- Su Shing Chen and Sidney A. Morris, Varieties of topological groups generated by Lie groups, Proc. Edinburgh Math. Soc. (2) 18 (1972/73), 49–53. MR 323958, DOI 10.1017/S0013091500026146
- Paul Civin and Bertram Yood, Quasi-reflexive spaces, Proc. Amer. Math. Soc. 8 (1957), 906–911. MR 90020, DOI 10.1090/S0002-9939-1957-0090020-6
- Mahlon M. Day, Normed linear spaces, Reihe: Reelle Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0094675
- Joseph Diestel, Sidney A. Morris, and Stephen A. Saxon, Varieties of locally convex topological vector spaces, Bull. Amer. Math. Soc. 77 (1971), 799–803. MR 282188, DOI 10.1090/S0002-9904-1971-12811-2 E. Dubinsky and M. S. Ramanujan, Power series nuclear spaces (in preparation).
- A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI 10.4153/cjm-1953-017-4
- Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
- John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205028
- M. Ĭ. Kadec′, Linear dimension of the spaces $L_{p}$ and $l_{q}$, Uspehi Mat. Nauk 13 (1958), no. 6 (84), 95–98 (Russian). MR 0101486
- M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/62), 161–176. MR 152879, DOI 10.4064/sm-21-2-161-176
- Shizuo Kakutani, Free topological groups and infinite direct product topological groups, Proc. Imp. Acad. Tokyo 20 (1944), 595–598. MR 14093
- N. J. Kalton, Unconditional and normalised bases, Studia Math. 38 (1970), 243–253 (errata insert). MR 276716, DOI 10.4064/sm-38-1-248-253
- Samuel Kaplan, Cartesian products of reals, Amer. J. Math. 74 (1952), 936–954. MR 54944, DOI 10.2307/2372236
- Takako K\B{o}mura and Yukio K\B{o}mura, Über die Einbettung der nuklearen Räume in $(s)^{A}$, Math. Ann. 162 (1965/66), 284–288 (German). MR 188754, DOI 10.1007/BF01360917
- Elton Lacey and R. J. Whitley, Conditions under which all the bounded linear maps are compact, Math. Ann. 158 (1965), 1–5. MR 173159, DOI 10.1007/BF01370391
- J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
- Robert H. Lohman, An embedding theorem for separable locally convex spaces, Canad. Math. Bull. 14 (1971), 114–120. MR 291756, DOI 10.4153/CMB-1971-023-1 A. A. Markov, On free topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 9 (1945), 3-64; English transl., Amer. Math. Soc. Transl. (1) 8 (1962), 195-272. MR 7, 7.
- André Martineau, Sur une propriété universelle de l’espace des distributions de M. Schwartz, C. R. Acad. Sci. Paris 259 (1964), 3162–3164 (French). MR 169051
- S. Mazur and W. Orlicz, Sur les espaces métriques linéaires. II, Studia Math. 13 (1953), 137–179 (French). MR 68730, DOI 10.4064/sm-13-2-137-179
- E. Michael, A short proof of the Arens-Eells embedding theorem, Proc. Amer. Math. Soc. 15 (1964), 415–416. MR 162222, DOI 10.1090/S0002-9939-1964-0162222-5
- A. A. Miljutin, Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 2 (1966), 150–156. (1 foldout) (Russian). MR 0206695
- Sidney A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145–160. MR 259010, DOI 10.1017/S0004972700041393
- Sidney A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145–160. MR 259010, DOI 10.1017/S0004972700041393
- Sidney A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145–160. MR 259010, DOI 10.1017/S0004972700041393
- Sidney A. Morris, Varieties of topological groups and left adjoint functors, J. Austral. Math. Soc. 16 (1973), 220–227. Collection of articles dedicated to the memory of Hanna Neumann, II. MR 0333059
- Sidney A. Morris, Free products of topological groups, Bull. Austral. Math. Soc. 4 (1971), 17–29. MR 274647, DOI 10.1017/S0004972700046219 —, Free compact abelian groups (submitted for publication). —, Varieties of topological groups, Bull. Austral. Math. Soc. 3 (1970), 429-431.
- Sidney A. Morris, Quotient groups of topological groups with no small subgroups, Proc. Amer. Math. Soc. 31 (1972), 625–626. MR 293000, DOI 10.1090/S0002-9939-1972-0293000-9
- Sidney A. Morris, Locally compact abelian groups and the variety of topological groups generated by the reals, Proc. Amer. Math. Soc. 34 (1972), 290–292. MR 294560, DOI 10.1090/S0002-9939-1972-0294560-4
- Sidney A. Morris, A topological group characterization of those locally convex spaces having their weak topology, Math. Ann. 195 (1972), 330–331. MR 295034, DOI 10.1007/BF01423619
- Hanna Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967. MR 0215899
- R. E. A. C. Paley, Some theorems on abstract spaces, Bull. Amer. Math. Soc. 42 (1936), no. 4, 235–240. MR 1563277, DOI 10.1090/S0002-9904-1936-06277-4 N. T. Peck, Every locally convex space is a subspace of a nearly exotic space (to appear).
- A. Pełczyński, On the isomorphism of the spaces $m$ and $M$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958), 695–696. MR 0102727
- A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR 126145, DOI 10.4064/sm-19-2-209-228
- A. Pełczyński, On Banach spaces containing $L_{1}(\mu )$, Studia Math. 30 (1968), 231–246. MR 232195, DOI 10.4064/sm-30-2-231-246
- A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968), 92. MR 227751
- A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces $C(\Omega )$ for $\Omega$ without perfect subsets, Studia Math. 18 (1959), 211–222. MR 107806, DOI 10.4064/sm-18-2-211-222
- E. S. Pondiczery, Power problems in abstract spaces, Duke Math. J. 11 (1944), 835–837. MR 11104
- M. S. Ramanujan, Power series spaces $\Lambda (\alpha )$ and associated $\Lambda (\alpha )$-nuclearity, Math. Ann. 189 (1970), 161–168. MR 270110, DOI 10.1007/BF01352441
- J. R. Retherford and C. W. McArthur, Some remarks on bases in linear topological spaces, Math. Ann. 164 (1966), 38–41. MR 194873, DOI 10.1007/BF01351809
- S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 471–473, XL (English, with Russian summary). MR 0088682
- Haskell P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from $L^{p}\,(\mu )$ to $L^{r}\,(\nu )$, J. Functional Analysis 4 (1969), 176–214. MR 0250036, DOI 10.1016/0022-1236(69)90011-1
- Stephen A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972), 87–106. MR 305010, DOI 10.1007/BF01419586
- Stephen A. Saxon, Embedding nuclear spaces in products of an arbitrary Banach space, Proc. Amer. Math. Soc. 34 (1972), 138–140. MR 318823, DOI 10.1090/S0002-9939-1972-0318823-9 —, (LF)-spaces, quasi-Baire spaces and the strongest locally convex topology (in preparation). —, Normalized Schauder bases in Fréchet spaces (in preparation).
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- W. J. Stiles, On properties of subspaces of $l_{p},\,0<p<1$, Trans. Amer. Math. Soc. 149 (1970), 405–415. MR 261315, DOI 10.1090/S0002-9947-1970-0261315-9
- W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53–61. MR 227743, DOI 10.4064/sm-30-1-53-61
- T. Terzioğlu, On Schwartz spaces, Math. Ann. 182 (1969), 236–242. MR 247417, DOI 10.1007/BF01350326
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 172 (1972), 207-230
- MSC: Primary 46A05; Secondary 46B99, 46M15
- DOI: https://doi.org/10.1090/S0002-9947-1972-0316992-2
- MathSciNet review: 0316992