Book Review
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MathSciNet review:
1567199
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Book Information:
Authors:
Charalambos D. Aliprantis and
Owen Burkinshaw
Title:
Locally solid Riesz spaces
Additional book information:
Academic Press, New York, 1978, xii + 198 pp.
Ichirô Amemiya, A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I. 12 (1953), 111–156. MR 0056853
Ichiro Amemiya, On ordered topological linear spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 14–23. MR 0138971
T. Andô, Contractive projections in $L_{p}$ spaces, Pacific J. Math. 17 (1966), 391–405. MR 192340
S. J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. (3) 15 (1965), 599–631. MR 182661, DOI 10.1112/plms/s3-15.1.599
S. J. Bernau, Lateral and Dedekind completion of Archimedean lattice groups, J. London Math. Soc. (2) 12 (1975/76), no. 3, 320–322. MR 401579, DOI 10.1112/jlms/s2-12.3.320
S. J. Bernau and H. Elton Lacey, The range of a contractive projection on an $L_{p}$-space, Pacific J. Math. 53 (1974), 21–41. MR 358325
S. J. Bernau and H. E. Lacey, Banach lattices and local unconditional structure, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 405–426. MR 606227
Roger D. Bleier, Free vector lattices, Trans. Amer. Math. Soc. 176 (1973), 73–87. MR 311541, DOI 10.1090/S0002-9947-1973-0311541-8
F. Bohnenblust, An axiomatic characterization of $L_p$-spaces, Duke Math. J. 6 (1940), 627–640. MR 2701
N. Bourbaki, Eléments de mathématique. XIII. Première partie: Les structures fondamentales de l’analyse. Livre VI: Intégration. Chapitre I: Inégalités de convexité. Chapitre II: Espaces de Riesz. Chapitre III: Mesures sur les espaces localement compacts. Chapitre IV: Prolongement d’une mesure; espaces $L^p$, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175, Hermann & Cie, Paris, 1952 (French). MR 0054691
R. G. Douglas, Contractive projections on an ${\mathfrak {L}}_{1}$ space, Pacific J. Math. 15 (1965), 443–462. MR 187087
D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. MR 0454575
13. D. H. Fremlin, Review of Banach lattices and positive operators by H. H. Schaefer, Bull. London Math. Soc. 8 (1976), 215-216.
14. H. Freudenthal, Teilweise geordnete Moduln, Nederl. Akad. Wetensch. Proc. Ser. A 39 (1936), 641-651.
A. Grothendieck, Une caractérisation vectorielle-métrique des espaces $L^1$, Canadian J. Math. 7 (1955), 552–561 (French). MR 76301, DOI 10.4153/CJM-1955-060-6
J. R. Isbell, A structure space for certain lattice-ordered groups and rings, J. London Math. Soc. 40 (1965), 63–71. MR 172940, DOI 10.1112/jlms/s1-40.1.63
Shizuo Kakutani, Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 4095, DOI 10.2307/1968915
Shizuo Kakutani, Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2) 42 (1941), 994–1024. MR 5778, DOI 10.2307/1968778
19. L. V. Kantorovich, Lineare halbgeordnete Raume, Mat. Sb. 44 (1937), 121-168.
H. Elton Lacey, Local unconditional structure in Banach spaces, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976) Lecture Notes in Math., Vol. 604, Springer, Berlin, 1977, pp. 44–56. MR 0454596
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
22. W. A. J. Luxemburg, Notes on Banach function spaces, Nederl. Akad. Wetensch. Proc. Ser. A, Note XIV 68 (1965), 229-248; Note XV 68 (1965), 415-446; Note XVI 68 (1965), 646-667.
23. W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach Junction spaces, Nederl. Akad. Wetensch. Proc. Ser. A, Note 1 66 (1963), 135-147; Note II 66 (1963), 148-153; Note III 66 (1963), 239-250; Note IV 66 (1963), 251-263; Note V 66 (1963), 496-504; Note VI 66 (1963), 655-668; Note VII 66 (1963), 669-681; Note VIII 67 (1964), 104-119; Note IX 67 (1964), 360-376; Note X 67 (1964), 493-506; Note XI 67 (1964), 507-518; Note XII 67 (1964), 519-529; Note XIII 67 (1964), 530-543.
24. W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. I, North-Holland, Amsterdam, 1971.
Fumitomo Maeda and Tôzirô Ogasawara, Representation of vector lattices, J. Sci. Hiroshima Univ. Ser. A 12 (1942), 17–35 (Japanese). MR 29087
Hidegorô Nakano, Über das System aller stetigen Funktionen auf einem topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), 308–310 (German). MR 14173
Hidegorô Nakano, Über normierte teilweisegeordnete Moduln, Proc. Imp. Acad. Tokyo 17 (1941), 311–317 (German). MR 14174
Hidegorô Nakano, Eine Spektraltheorie, Proc. Phys.-Math. Soc. Japan (3) 23 (1941), 485–511 (German). MR 5797
Hidegorô Nakano, Teilweise geordnete Algebra, Jpn. J. Math. 17 (1941), 425–511 (German). MR 5796, DOI 10.4099/jjm1924.17.0_{4}25
Hidegorô Nakano, Stetige lineare Funktionale auf dem teilweisegeordneten Modul, J. Fac. Sci. Imp. Univ. Tokyo Sect. I. 4 (1942), 201–382 (German). MR 0022308
Hidegorô Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950. MR 0038565
Hidegorô Nakano, Linear topologies on semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I. 12 (1953), 87–104. MR 0056851
Tôzirô Ogasawara, Theory of vector lattices. I, J. Sci. Hiroshima Univ. Ser. A 12 (1942), 37–100 (Japanese). MR 29089
34. F. Riesz, Sur la décomposition des opérations fonctionelles linéaires, Atti del Congr. Internaz. dei Mat., Bologna 1928, 3 (1930), 143-148; Oeuvres Completes II, 1097-1102, Budapest, 1960.
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
František Šik, Zur Theorie der halbgeordneten Gruppen, Czechoslovak Math. J. 6(81) (1956), 1–25 (Russian, with German summary). MR 81907
37. S. W. P. Steen, An introduction to the theory of operators. I, Proc. London Math. Soc. (2) 41 (1936), 361-392.
M. H. Stone, A general theory of spectra. I, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 280–283. MR 2023, DOI 10.1073/pnas.26.4.280
L. Tzafriri, Remarks on contractive projections in $L_{p}$-spaces, Israel J. Math. 7 (1969), 9–15. MR 248514, DOI 10.1007/BF02771741
Kôsaku Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 17 (1941), 121–124. MR 5795
Kôsaku Yosida, Vector lattices and additive set functions, Proc. Imp. Acad. Tokyo 17 (1941), 228–232. MR 8119
Kôsaku Yosida, On the representation of the vector lattice, Proc. Imp. Acad. Tokyo 18 (1942), 339–342. MR 15378
- 1.
- I. Amemiya, A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I, 12 (1953), 111-156. MR 0056853
- 2.
- I. Amemiya, On ordered topological linear spaces, Proc. Internat. Sympos. Linear Spaces, Jerusalem, (1960) pp. 14-23, Israel Acad. Sci. Humanities, Jerusalem, 1961. MR 138971
- 3.
- T. Ando, Contractive projections in L, Pacific J. Math. 17 (1966), 391-405. MR 192340
- 4.
- S. J. Bernau, Unique representation of archimedean lattice groups and normal archimedean lattice rings, Proc. London Math. Soc. (3) 15 (1965), 599-631. MR 182661
- 5.
- S. J. Bernau, Lateral and Dedekind completion of Archimedean lattice groups, J. London Math. Soc. 12 (1976), 320-322. MR 401579
- 6.
- S. J. Bernau and H. Elton Lacey, The range of a contractive projection on an L, Pacific J. Math. 53 (1974), 21-41. MR 358325
- 7.
- S. J. Bernau and H. Elton Lacey, Banach lattices and local unconditional structure, Seminar in Analysis, Lecture Notes in Math., Springer-Verlag, Berlin and New York (to appear). MR 606227
- 8.
- R. D. Bleier, Free vector lattices, Trans. Amer. Math. Soc. 176 (1973), 73-87. MR 311541
- 9.
- H. F. Bohnenblust, On axiomatic characterization of L, Duke Math. J. 6 (1940), 627-640. MR 2701
- 10.
- N. Bourbaki, Intégration, Chapitre II, Actualités Sci. Indust. no. 1175, Hermann et Cie, Paris, 1952. MR 54691
- 11.
- R. G. Douglas, Contractive projections on an L, Pacific J. Math. 15 (1965), 443-462. MR 187087
- 12.
- D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge Univ. Press, London and New York, 1974. MR 454575
- 13.
- D. H. Fremlin, Review of Banach lattices and positive operators by H. H. Schaefer, Bull. London Math. Soc. 8 (1976), 215-216.
- 14.
- H. Freudenthal, Teilweise geordnete Moduln, Nederl. Akad. Wetensch. Proc. Ser. A 39 (1936), 641-651.
- 15.
- A. Grothendieck, Une characterisation vectorielle metrique des espaces L, Canad. J. Math. 7 (1955), 552-561. MR 76301
- 16.
- J. R. Isbell, A structure space for certain lattice-ordered groups and rings, J. London Math. Soc. 40 (1965), 63-71. MR 172940
- 17.
- S. Kakutani, Concrete representation of abstract (L)-spaces and the mean ergodic theorem, Ann. of Math. 42 (1941), 523-537. MR 4095
- 18.
- S. Kakutani, Concrete representations of abstract (M)-spaces, Ann, of Math. 42 (1941), 994-1024. MR 5778
- 19.
- L. V. Kantorovich, Lineare halbgeordnete Raume, Mat. Sb. 44 (1937), 121-168.
- 20.
- H. Elton Lacey, Local unconditional structure in Banach spaces, in Banach spaces of analytic functions (Proc. Pelczyński Conf., Kent State Univ., Kent, Ohio 1976) pp. 44-56. Lecture Notes in Math., vol. 604, Springer-Verlag, Berlin and New York, 1977. MR 454596
- 21.
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, volume II: Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, Heidelberg, New York, 1979. MR 540367
- 22.
- W. A. J. Luxemburg, Notes on Banach function spaces, Nederl. Akad. Wetensch. Proc. Ser. A, Note XIV 68 (1965), 229-248; Note XV 68 (1965), 415-446; Note XVI 68 (1965), 646-667.
- 23.
- W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach Junction spaces, Nederl. Akad. Wetensch. Proc. Ser. A, Note 1 66 (1963), 135-147; Note II 66 (1963), 148-153; Note III 66 (1963), 239-250; Note IV 66 (1963), 251-263; Note V 66 (1963), 496-504; Note VI 66 (1963), 655-668; Note VII 66 (1963), 669-681; Note VIII 67 (1964), 104-119; Note IX 67 (1964), 360-376; Note X 67 (1964), 493-506; Note XI 67 (1964), 507-518; Note XII 67 (1964), 519-529; Note XIII 67 (1964), 530-543.
- 24.
- W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. I, North-Holland, Amsterdam, 1971.
- 25.
- F. Maeda and T. Ogasawara, Representation of vector lattices, J. Sci. Hiroshima Univ. A, 12 (1942), 17-35. MR 29087
- 26.
- H. Nakano, Über das System aller stetigen Funktionen auf einem topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1940-41), 308-310. MR 14173
- 27.
- H. Nakano, Uber normierte, teilweise geordnete Moduln, Proc. Imp. Acad. Tokyo 17 (1940-41), 311-317. MR 14174
- 28.
- H. Nakano, Eine Spektraltheorie, Proc. Phys.-Math. Soc. Japan 23 (1941), 485-511. MR 5797
- 29.
- H. Nakano, Teilweise geordnete Algebra, Japan J. Math. 17 (1941), 425-511. MR 5796
- 30.
- H. Nakano, Stetige lineare Funktionale auf dem teilweise-geordnetem Modul, J. Fac. Sci. Imp. Univ. Tokyo 4 (1942), 201-382. MR 22308
- 31.
- H. Nakano, Modulared semi-ordered linear spaces, Tokyo, 1950. MR 38565
- 32.
- H. Nakano, Linear topologies on semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. (1) 12 (1953), 87-104. MR 56851
- 33.
- T. Ogasawara, Theory of vector lattices. I and II, J. Sci. Hiroshima Univ. (A) 12 (1942), 37-100 and 13 (1944), 41-161 (in Japanese). MR 29089
- 34.
- F. Riesz, Sur la décomposition des opérations fonctionelles linéaires, Atti del Congr. Internaz. dei Mat., Bologna 1928, 3 (1930), 143-148; Oeuvres Completes II, 1097-1102, Budapest, 1960.
- 35.
- H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Heidelberg, 1974. MR 423039
- 36.
- F. Sik, Zur Theorie der halbgeordneten Gruppen, (Russian, German Summary), Czech, Math. J. 81 (1956), 1-25. MR 81907
- 37.
- S. W. P. Steen, An introduction to the theory of operators. I, Proc. London Math. Soc. (2) 41 (1936), 361-392.
- 38.
- M. H. Stone, A general theory of spectra. I and II, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 280-283 and 27 (1941), 83-87. MR 2023
- 39.
- L. Tzafriri, Remarks on contractive projections in L, Israel J. Math., 7 (1969), 9-15. MR 248514
- 40.
- K. Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 17 (1940-41), 121-124. MR 5795
- 41.
- K. Yosida, Vector lattices and additive set functions, Proc. Imp. Acad. Tokyo 17 (1940-41), 228-232. MR 8119
- 42.
- K. Yosida, On the representation of the vector lattice, Proc. Imp. Acad. Tokyo 18 (1941-42), 339-342. MR 15378
Review Information:
Reviewer:
Simon J. Bernau
Journal:
Bull. Amer. Math. Soc.
1 (1979), 997-1003
DOI:
https://doi.org/10.1090/S0273-0979-1979-14715-3