Book Review
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MathSciNet review:
1566874
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Book Information:
Author:
A. Grothendieck
Title:
Topological vector spaces
Additional book information:
Gordon and Breach, New York, London, Paris, 1973, x + 245 pp., $19.50. (Translated by Orlando Chaljub).
1. J. Batt, P. Dierolf and J. Voigt, Summable sequences and topological properties of m (to appear).
S. R. Caradus, W. E. Pfaffenberger, and Bertram Yood, Calkin algebras and algebras of operators on Banach spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 9, Marcel Dekker, Inc., New York, 1974. MR 0415345
Marc De Wilde, Réseaux dans les espaces linéaires à semi-normes, Mém. Soc. Roy. Sci. Liège Coll. in-8$^\circ$ (5) 18 (1969), no. 2, 144 (French). MR 285885
M. De Wilde, Various types of barrelledness and increasing sequences of balanced and convex sets in locally convex spaces, Summer School on Topological Vector Spaces (Univ. Libre Bruxelles, Brussels, 1972) Lecture Notes in Math., Vol. 331, Springer, Berlin, 1973, pp. 211–217. MR 0482031
M. De Wilde and C. Houet, On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann. 192 (1971), 257–261. MR 288549, DOI 10.1007/BF02075355
M. A. Dostal, Some recent results on topological vector spaces, Functional analysis and applications (Proc. Sympos. Analysis, Univ. Fed. Pernambuco, Recife, 1972) Lecture Notes in Math., Vol. 384, Springer, Berlin, 1974, pp. 20–91. MR 0397437
Adrien Douady, Parties compactes d’un espace de fonctions continues à support compact, C. R. Acad. Sci. Paris 257 (1963), 2788–2791 (French). MR 156190
8. V. Eberhardt, Durch Graphensätze definierte lokalkonvexe Räume, Thesis, München, 1972.
Volker Eberhardt, Über einen Graphensatz für Abbildungen mit normiertem Zielraum, Manuscripta Math. 12 (1974), 47–65 (German, with English summary). MR 336280, DOI 10.1007/BF01166233
V. Eberhardt, Einige Vererbbarkeitseigenschaften von $B$- und $B_{r}$ -vollständigen Räumen, Math. Ann. 215 (1975), 1–11 (German). MR 370119, DOI 10.1007/BF01351786
Volker Eberhardt and Walter Roelcke, Über einen Graphensatz für lineare Abbildungen mit metrisierbarem Zielraum, Manuscripta Math. 13 (1974), 53–68 (German, with English summary). MR 355517, DOI 10.1007/BF01168742
Benno Fuchssteiner, Sandwich theorems and lattice semigroups, J. Functional Analysis 16 (1974), 1–14. MR 0345889, DOI 10.1016/0022-1236(74)90068-8
13. G. B. Giráldez Tiebo, Teoremas de gráfico cerrado, nucleo cerrado y aplicación abierta, Thesis, Madrid, 1974.
A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168–186 (French). MR 47313, DOI 10.2307/2372076
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, Sur les espaces ($F$) et ($DF$), Summa Brasil. Math. 3 (1954), 57–123 (French). MR 75542
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Charles Harvey, On domination estimates and global existence, J. Math. Mech. 16 (1967), 675–702. MR 0214903
Jean Horváth, Quelques résultats de M. H. Powell sur le théorème du graphe fermé, Publ. Dép. Math. (Lyon) 10 (1973), no. 3, 219–231 (French). MR 365077
Taqdir Husain, Two new classes of locally convex spaces, Math. Ann. 166 (1966), 289–299. MR 212535, DOI 10.1007/BF01360784
Hans Jarchow, Nuclear locally convex spaces, University of Maryland, Department of Mathematics, College Park, Md., 1976. Department of Mathematics, University of Maryland, Lecture Note, No. 13. MR 0461080
22. P. Jimenez Guerra, Thesis, Madrid, 1974.
Gottfried Köthe, General linear transformations of locally convex spaces, Math. Ann. 159 (1965), 309–328. MR 192347, DOI 10.1007/BF01362768
Stephen Saxon and Mark Levin, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91–96. MR 280972, DOI 10.1090/S0002-9939-1971-0280972-0
A. Marquina and P. Pérez Carreras, On quasibarrelled spaces, Manuscripta Math. 12 (1974), 387–398. MR 412760, DOI 10.1007/BF01171082
P. Pérez Carreras and A. Marquina, Other results in ${\cal A}{\cal B}$-barrelled spaces, Proceedings of the Symposium on Functional Analysis (Istanbul, 1973), Publ. Math. Res. Inst.-Istanbul, No. 1, Math. Res. Inst., Istanbul, 1974, pp. 115–116. MR 0420190
Dan Randtke, Characterization of precompact maps, Schwartz spaces and nuclear spaces, Trans. Amer. Math. Soc. 165 (1972), 87–101. MR 305009, DOI 10.1090/S0002-9947-1972-0305009-1
Wendy Robertson, On the closed graph theorem and spaces with webs, Proc. London Math. Soc. (3) 24 (1972), 692–738. MR 306857, DOI 10.1112/plms/s3-24.4.692
Wendy Robertson, Sur le théorème du graphe fermé, Actes du Colloque d’Analyse Fonctionnelle (Univ. Bordeaux, Bordeaux, 1971), Bull. Soc. Math. France Mém., No. 31-32, Soc. Math. France, Paris, 1972, pp. 343–350 (French). MR 0385508, DOI 10.24033/msmf.99
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, Some normed barrelled spaces which are not Baire, Math. Ann. 209 (1974), 153–160. MR 352925, DOI 10.1007/BF01351318
Aaron R. Todd and Stephen A. Saxon, A property of locally convex Baire spaces, Math. Ann. 206 (1973), 23–34. MR 328523, DOI 10.1007/BF01431526
33. L. Scheller, Fortsetzungen additiver Funktionen auf Halbgruppen zwischen sub- und superadditiven Funktionalen, Thesis, Frankfurt am Main, 1973.
Laurent Schwartz, Homomorphismes et applications complètement continues, C. R. Acad. Sci. Paris 236 (1953), 2472–2473 (French). MR 57457
François Trèves, Locally convex spaces and linear partial differential equations, Die Grundlehren der mathematischen Wissenschaften, Band 146, Springer-Verlag New York, Inc., New York, 1967. MR 0223939
F. Treves, Applications of distributions to $\textrm {PDE}$ theory, Amer. Math. Monthly 77 (1970), 241–248. MR 252818, DOI 10.2307/2317707
Manuel Valdivia, A hereditary property in locally convex spaces, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 2, 1–2 (English, with French summary). MR 333642
Manuel Valdivia, Absolutely convex sets in barrelled spaces, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 2, 3–13 (English, with French summary). MR 333643
Manuel Valdivia, On final topologies, J. Reine Angew. Math. 251 (1971), 193–199. MR 295040, DOI 10.1515/crll.1971.251.193
Manuel Valdivia, A class of bornological barrelled spaces which are not ultrabornological, Math. Ann. 194 (1971), 43–51. MR 313753, DOI 10.1007/BF01351821
Manuel Valdivia, Some examples on quasi-barrelled spaces, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 2, 21–26 (English, with French summary). MR 336277
Manuel Valdivia, On nonbornological barrelled spaces, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 2, 27–30 (English, with French summary). MR 336274
- 1.
- J. Batt, P. Dierolf and J. Voigt, Summable sequences and topological properties of m (to appear).
- 2.
- S. R. Caradus, W. E. Pfaffenberger and B. Yood, Calkin algebras and algebras of operators on Banach spaces, Lecture Notes in Pure and Appl. Math., vol. 9, Dekker, New York, 1974. MR 0415345
- 3.
- M. De Wilde, Réseaux dans les espaces linéaires à semi-normes, Mém. Soc. Roy. Sci. Liège Coll. in-8° (5) 18 (1969), no. 2, 144 pp. MR 44 #3102. MR 285885
- 4.
- M. De Wilde, Various types of barrelledness and increasing sequences of balanced and convex sets in locally convex spaces, Lecture Notes in Math., vol. 331, Springer-Verlag, Berlin, Heidelberg, New York, 1973, pp. 211-217. MR 482031
- 5.
- M. De Wilde and C. Houet, On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann. 192 (1971), 257-261. MR 44 #5746. MR 288549
- 6.
- M. Dostal, Some recent results on topological vector spaces, Lecture Notes in Math., vol. 384, Springer-Verlag, Berlin, Heidelberg, New York, 1974, pp. 21-97. MR 397437
- 7.
- A. Douady, Parties compacte d'un espace de fonctions continues à support compact, C. R. Acad. Sci. Paris 257 (1963), 2788-2791. MR 27 #6121. MR 156190
- 8.
- V. Eberhardt, Durch Graphensätze definierte lokalkonvexe Räume, Thesis, München, 1972.
- 9.
- V. Eberhardt, Über einen Graphensatz für Abbildungen mit normiertem Zielraum, Manuscripta Math. 12 (1974), 47-65. MR 49 #1056. MR 336280
- 10.
- V. Eberhardt, Einige Vererbbarkeitseigenschaften von B- und B, Math. Ann. 215 (1975), 1-11. MR 370119
- 11.
- V. Eberhardt and W. Roelcke, Über einen Graphensatz für lineare Abbildungen mit metrisierbarem Zielraum, Manuscripta Math. 13 (1974), 53-68. MR 355517
- 12.
- B. Fuchssteiner, Sandwich theorems and lattice semigroups, J. Functional Analysis 16 (1974), 1-14. MR 49 #10619. MR 345889
- 13.
- G. B. Giráldez Tiebo, Teoremas de gráfico cerrado, nucleo cerrado y aplicación abierta, Thesis, Madrid, 1974.
- 14.
- A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168-186. MR 13, 857. MR 47313
- 15.
- A. Grothendieck, Sur les applications linéaires failblement compactes d'espaces du type C(K), Canad. J. Math. 5 (1953), 129-173. MR 15, 438. MR 58866
- 16.
- A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasil. Math. 3 (1954), 57-123. MR 17, 765. MR 75542
- 17.
- A. Grothendieck, Produits tensoriels topologiques et espaces nucléaries, Mem. Amer. Math. Soc. No. 16 (1955). MR 17, 763. MR 75539
- 18.
- C. Harvey, On domination estimates and global existence, J. Math. Mech. 16 (1967), 675-702. MR 35 #5750. MR 214903
- 19.
- J. Horváth, Quelques résultats de M. H. Powell sur le théorème du graphe fermé, Publ. Dép. Math. (Lyon) 10 (1973), 65-77. MR 51 # 1330. MR 365077
- 20.
- T. Husain, Two new classes of locally convex spaces, Math. Ann. 166 (1966), 289-299. MR 35 #3406. MR 212535
- 21.
- H. Jarchow, Nuclear locally convex spaces, Lecture notes, no. 13, University of Maryland, 1976. MR 461080
- 22.
- P. Jimenez Guerra, Thesis, Madrid, 1974.
- 23.
- G. Köthe, General linear transformations of locally convex spaces, Math. Ann. 159 (1965), 309-328. MR 33 #573. MR 192347
- 24.
- M. Levin and S. Saxon, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91-96. MR 43 #6691. MR 280972
- 25.
- A. Marquina and P. Pérez Carreras, On quasibarrelled spaces, Manuscripta Math. 12 (1974), 387-398. MR 412760
- 26.
- A. Marquina and P. Pérez Carreras, Other results in PB-barrelled spaces, Proc. Sympos. Functional Analysis, Istanbul, Silivri, 1973, pp. 115-116. MR 420190
- 27.
- D. J. Randtke, Characterization of precompact maps, Schwartz spaces and nuclear spaces, Trans. Amer. Math. Soc. 165 (1972), 87-101. MR 46 #4139. MR 305009
- 28.
- W. Robertson, On the closed graph theorem and spaces with webs, Proc. London Math. Soc. (3) 24 (1972), 692-738. MR 46 #5979. MR 306857
- 29.
- W. Robertson, Sur le théorème du graphe fermé, Bull. Soc. Math. France Suppl. Mém. 31/32 (1972), 343-350. MR 385508
- 30.
- S. A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972), 87-106. MR 46 #4140. MR 305010
- 31.
- S. A. Saxon, Some normed barrelled spaces which are not Baire, Math. Ann. 209 (1974), 153-160. MR 352925
- 32.
- S. A. Saxon and A. R. Todd, A property of locally convex Baire spaces, Math. Ann. 206 (1973), 23-34. MR 48 #6865. MR 328523
- 33.
- L. Scheller, Fortsetzungen additiver Funktionen auf Halbgruppen zwischen sub- und superadditiven Funktionalen, Thesis, Frankfurt am Main, 1973.
- 34.
- L. Schwartz, Homomorphismes et applications complètement continues, C. R. Acad. Sci. Paris 236 (1953), 2472-2473. MR 15, 233. MR 57457
- 35.
- F. Trèves, Locally convex spaces and linear partial differential equations Die Grundlehren der math. Wissenschaften, Band 146, Springer-Verlag, New York, 1967. MR 36 #6986. MR 223939
- 36.
- F. Trèves, Applications of distributions to PDE theory, Amer. Math. Monthly 77 (1970), 241-248. MR 40 #6035. MR 252818
- 37.
- M. Valdívia, A hereditary property in locally convex spaces, Ann. Inst. Fourier (Grenoble) 21 (1972), fasc. 2, 1-2. MR 48 #11967. MR 333642
- 38.
- M. Valdívia, Absolutely convex sets in barrelled spaces, Ann. Inst. Fourier (Grenoble) 21 (1972), fasc. 2, 3-13. MR 48 #11968. MR 333643
- 39.
- M. Valdívia, On final topologies, J. Reine Angew. Math. 251 (1971), 193-199. MR 45 #4108. MR 295040
- 40.
- M. Valdívia, A class of bornological barrelled spaces which are not ultrabornological, Math. Ann. 194 (1971), 43-51. MR 47 #2307. MR 313753
- 41.
- M. Valdívia, Some examples on quasi-barrelled spaces, Ann. Inst. Fourier (Grenoble) 22 (1972), fasc. 2, 21-26. MR 49 # 1053. MR 336277
- 42.
- M. Valdívia, On nonbornological barrelled spaces, Ann. Inst. Fourier (Grenoble) 22 (1972), fasc. 2, 27-30. MR 49 # 1050. MR 336274
Review Information:
Reviewer:
John Horváth
Journal:
Bull. Amer. Math. Soc.
82 (1976), 515-521
DOI:
https://doi.org/10.1090/S0002-9904-1976-14076-1
