نگاهی به ابرپالایه‌ها در ریاضیات

نوع مقاله : مقاله مروری

نویسندگان

1 دانش‌آموختهٔ دکترای ریاضی، دانشگاه فردوسی مشهد

2 دانشگاه فردوسی مشهد، دانشکدهٔ علوم ریاضی، گروه ریاضی محض

چکیده

ابرپالایه مفهومی کارآمد در سراسر ریاضیات است. در این مقاله، با تکیه بر ویژگی‌ها و کاربردهای ابرپالایه سعی در شناختِ ساختار این مفهوم داریم. به‌این‌منظور با بررسی نمودهای مختلف ابرپالایه در نظریۀ مجموعه‌ها، توپولوژی، نظریۀ اندازه‌، فشرده‌سازی استون-چخ، آنالیز نااستاندارد، و به‌ویژه آنالیز تابعی به توصیف ابرپالایه می‌پردازیم.

کلیدواژه‌ها

موضوعات


[1]‎Arens‎, ‎R.‎, The adjoint of a bilinear operation‎,

‎ Proc‎. ‎Amer‎. ‎Math‎. ‎Soc., ‎2 (1951)‎, ‎839-848‎.

[2]‎ Bell‎, ‎J‎. ‎L.‎, ‎Fremlin‎, ‎D‎. ‎H.‎,

‎ A geometric form of the axiom of choice‎,

‎ Fund‎. ‎Math., 77 (1972)‎, ‎167-170‎.

[3]‎Cartan‎, ‎H.‎,‎ Filtres et ultrafiltres‎,

‎C‎. ‎R‎. ‎Acad‎. ‎Sci‎. ‎Paris, ‎205 (1937)‎, ‎777-779‎.

[4]‎Chang‎, ‎C‎. ‎C.‎, ‎Keisler‎, ‎H‎. ‎J.‎,

‎Model Theory, ‎3rd‎. ‎ed.‎, ‎North-Holland Publishing Co.‎, ‎Amsterdam‎, ‎1990‎ .

[5]‎Dales‎, ‎H‎. ‎G.‎,‎Banach Algebras and Automatic Continuity,

‎ Clarendon Press‎, ‎Oxford‎, ‎2000‎.

[6]‎Daws‎, ‎M.‎, ‎ Ultrapowers of Banach algebras and modules‎,

‎ Glasg‎. ‎Math‎. ‎J., ‎50 (2008)‎, ‎539-555‎.

[7]‎Dineen‎, ‎S.‎,‎ Complete holomorphic vector fields on the second dual of a Banach space‎,

‎ Math‎. ‎Scand., ‎59 (1986)‎, ‎131-142‎.

[8]‎Faal‎, ‎R.‎,‎ The Second Dual of the Algebra of Bounded Operators and Tensor Products‎,

‎Master`s thesis‎, ‎Ferdowsi University of Mashhad‎, ‎2013‎.

[9]‎Faal‎, ‎R.‎, ‎Ebrahimi Vishki‎, ‎H‎. ‎R.‎,‎ More on the Arens regularity of B(X)‎,

‎ Bull‎. ‎Austral‎. ‎Math‎. ‎Soc., 94 (2016)‎, ‎296-303‎.

[10]Faal, R., Ebrahimi Vishki, H. R., Dean’s identity and the principal of local reflexivity,

[11]Godefroy, G., Iochum, B., Arens-regularity of Banach algebras and the geometry of Banach spaces
J. Funct. Anal., 80 (1988), 47–59.

[12]Goldblatt, R., Lectures on the Hyperreals; an Introduction to Nonstandard Analysis, Springer-
Verlag, Graduate Texts in Mathematics, vol.188, Springer-Verlag, Berlin, 1998.

[13]Hamilton, A. G., Logic for Mathematicians, Cambridge University Press, Cambridge, 1988

[14]Harlpern,J. D., Levy, A., The Boolean prime ideal theorem does not imply the axiom of choice,
axiomatic set theory part 1, Proc. Symp. Pure Math., 13 (1971), 83–134.

[16]Hindman, N., Strauss, D., Algebra in the Stone-C￿ech Compactification- Theory and Applications,
Walter de Gruyter, Berlin, 2012.

[17]Iochum, B., Loupias, G., Remarks on the bidual of Banach algebra (the C* case), Annales scientifiques
de l’Universitй de Clermont-Ferrand, sйrie Mathйmatiques, 27 (1991), 107-118.

[18]Iochum, B., Loupias, G., Arens regularity and local reflexivity principle for Banach algebras,
Math. Ann., 284 (1989), 23–40.

[19]Jech, T. J., The Axiom of Choice, Dover Books on Mathematics, Dover, New York, 2008.

[20]Kirman, A., Sondermann, D., Arrow’s theorem, many agents, and invisible dictators, J. Econ.
Theory, 5 (1972), 267–277.

[21]Khosravi, A. A., Ebrahimi Vishki, H. R., Peralta, M., Aron-Berner extensions of triple maps with
applications to the bidual of Jordan Banach triple system.

[22]Łoś, J., Ryll-Nardzewski, C., On the application of Tychonoff’s theorem in mathematical proofs,
Studia Math., 38 (1951) 233–237.

[23]Luxemburg, W. A. J., Two applications of the method of construction by ultrapowers to analysis,
Bull. Amer. Math. Soc., 68 (1962), 416–419.

[24]Luxemburg, W. A. J., Reduced powers of the real number system and equivalents of the Hahn-
Banach extension theorem, in Appl. Model Theory Algebra, Anal., Probab., Proc. Int. Sympos.
Calif. Inst. Technol., 1969, 123–137.

[25]Martínez-Abejón, A., An elementary proof of the principle of local reflexivity, Proc. Amer. Math.
Soc., 127 (1999), 1397–1398.

[26]Pincus, D., Independence of the prime ideal theorem from the Hahn Banach theorem, Bull. Amer.
Math. Soc., 78 (1972), 766–770.

[27]Pincus, D., The strength of the Hahn-Banach theorem, Lecture Notes in Mathematics, vol. 369,
Springer-Verlag, Berlin, 1974, 203–248.

[28]Tao, T., Hilbert’s Fifth Problem and Related Topics, American Mathematical Society, Providence
RI, 2014.

[29]Vaught, L. R., Alfred Tarski’s work in model theory J. Symbolic Logic, 51 (1986), 869–882.

[30]Wang‎, ‎H.‎,‎A logical Journey‎: ‎From Godel to Philosophy,

‎ Bradford Book‎, ‎Massachusetts‎, ‎1977‎.

[1]‎Arens‎, ‎R.‎, The adjoint of a bilinear operation‎,
‎ Proc‎. ‎Amer‎. ‎Math‎. ‎Soc., ‎2 (1951)‎, ‎839-848‎.
[2]‎Bell‎, ‎J‎. ‎L.‎, ‎Fremlin‎, ‎D‎. ‎H.‎,
‎ A geometric form of the axiom of choice‎,
‎ Fund‎. ‎Math., 77 (1972)‎, ‎167-170‎.
[3]‎Cartan‎, ‎H.‎,
‎ Filtres et ultrafiltres‎,
‎C‎. ‎R‎. ‎Acad‎. ‎Sci‎. ‎Paris, ‎205 (1937)‎, ‎777-779‎.
[4]‎Chang‎, ‎C‎. ‎C.‎, ‎Keisler‎, ‎H‎. ‎J.‎,
‎Model Theory, ‎3rd‎. ‎ed.‎,
‎North-Holland Publishing Co.‎, ‎Amsterdam‎, ‎1990‎ .
[5]‎Dales‎, ‎H‎. ‎G.‎,‎Banach Algebras and Automatic Continuity,
‎ Clarendon Press‎, ‎Oxford‎, ‎2000‎.
[6]‎Daws‎, ‎M.‎,
‎ Ultrapowers of Banach algebras and modules‎,
‎ Glasg‎. ‎Math‎. ‎J., ‎50 (2008)‎, ‎539-555‎.
[7]‎Dineen‎, ‎S.‎,‎ Complete holomorphic vector fields on the second dual of a Banach space‎,
‎ Math‎. ‎Scand., ‎59 (1986)‎, ‎131-142‎.
[8]‎Faal‎, ‎R.‎,‎ The Second Dual of the Algebra of Bounded Operators and Tensor Products‎,
‎Master`s thesis‎, ‎Ferdowsi University of Mashhad‎, ‎2013‎.
[9]‎Faal‎, ‎R.‎, ‎Ebrahimi Vishki‎, ‎H‎. ‎R.‎,‎ More on the Arens regularity of B(X)‎,
‎ Bull‎. ‎Austral‎. ‎Math‎. ‎Soc., 94 (2016)‎, ‎296-303‎.
[10]Faal, R., Ebrahimi Vishki, H. R., Dean’s identity and the principal of local reflexivity,
[11]Godefroy, G., Iochum, B., Arens-regularity of Banach algebras and the geometry of Banach spaces
J. Funct. Anal., 80 (1988), 47–59.
[12]Goldblatt, R., Lectures on the Hyperreals; an Introduction to Nonstandard Analysis, Springer-
Verlag, Graduate Texts in Mathematics, vol.188, Springer-Verlag, Berlin, 1998.
[13]Hamilton, A. G., Logic for Mathematicians, Cambridge University Press, Cambridge, 1988
[14]Harlpern,J. D., Levy, A., The Boolean prime ideal theorem does not imply the axiom of choice,
axiomatic set theory part 1, Proc. Symp. Pure Math., 13 (1971), 83–134.
[16]Hindman, N., Strauss, D., Algebra in the Stone-C￿ech Compactification- Theory and Applications,
Walter de Gruyter, Berlin, 2012.
[17]Iochum, B., Loupias, G., Remarks on the bidual of Banach algebra (the C* case), Annales scientifiques
de l’Universitй de Clermont-Ferrand, sйrie Mathйmatiques, 27 (1991), 107-118.
[18]Iochum, B., Loupias, G., Arens regularity and local reflexivity principle for Banach algebras,
Math. Ann., 284 (1989), 23–40.
[19]Jech, T. J., The Axiom of Choice, Dover Books on Mathematics, Dover, New York, 2008.
[20]Kirman, A., Sondermann, D., Arrow’s theorem, many agents, and invisible dictators, J. Econ.
Theory, 5 (1972), 267–277.
[21]Khosravi, A. A., Ebrahimi Vishki, H. R., Peralta, M., Aron-Berner extensions of triple maps with
applications to the bidual of Jordan Banach triple system.
[22]Łoś, J., Ryll-Nardzewski, C., On the application of Tychonoff’s theorem in mathematical proofs,
Studia Math., 38 (1951) 233–237.
[23]Luxemburg, W. A. J., Two applications of the method of construction by ultrapowers to analysis,
Bull. Amer. Math. Soc., 68 (1962), 416–419.
[24]Luxemburg, W. A. J., Reduced powers of the real number system and equivalents of the Hahn-
Banach extension theorem, in Appl. Model Theory Algebra, Anal., Probab., Proc. Int. Sympos.
Calif. Inst. Technol., 1969, 123–137.
[25]Martínez-Abejón, A., An elementary proof of the principle of local reflexivity, Proc. Amer. Math.
Soc., 127 (1999), 1397–1398.
[26]Pincus, D., Independence of the prime ideal theorem from the Hahn Banach theorem, Bull. Amer.
Math. Soc., 78 (1972), 766–770.
[27]Pincus, D., The strength of the Hahn-Banach theorem, Lecture Notes in Mathematics, vol. 369,
Springer-Verlag, Berlin, 1974, 203–248.
[28]Tao, T., Hilbert’s Fifth Problem and Related Topics, American Mathematical Society, Providence
RI, 2014.
[29]Vaught, L. R., Alfred Tarski’s work in model theory J. Symbolic Logic, 51 (1986), 869–882.
[30]Wang‎, ‎H.‎,‎A logical Journey‎: ‎From Godel to Philosophy,
‎ Bradford Book‎, ‎Massachusetts‎, ‎1977‎.