Geodesic
Biequivalence vector spaces in the alternative set theory
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 3, pp. 517-544 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

As a counterpart to classical topological vector spaces in the alternative set theory, biequivalence vector spaces (over the field $Q$ of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view towards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total convexity of the monad and/or of the galaxy of $0$. Finally, the existence of a rather strong type of basis for a fairly extensive area of biequivalence vector spaces, containing all the most important particular cases, is established.
As a counterpart to classical topological vector spaces in the alternative set theory, biequivalence vector spaces (over the field $Q$ of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view towards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total convexity of the monad and/or of the galaxy of $0$. Finally, the existence of a rather strong type of basis for a fairly extensive area of biequivalence vector spaces, containing all the most important particular cases, is established.
Classification : 03E70, 03H05, 46A04, 46A06, 46A08, 46A09, 46A35, 46Q05, 46S20
Keywords: alternative set theory; biequivalence; vector space; monad; galaxy; symmetric Sd-closure; dual; valuation; norm; convex; basis 
@article{CMUC_1991_32_3_a12,
     author = {\v{S}m{\'\i}d, Miroslav and Zlato\v{s}, Pavol},
     title = {Biequivalence vector spaces in the alternative set theory},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {517--544},
     year = {1991},
     volume = {32},
     number = {3},
     mrnumber = {1159799},
     zbl = {0756.03027},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1991_32_3_a12/}
}
TY  - JOUR
AU  - Šmíd, Miroslav
AU  - Zlatoš, Pavol
TI  - Biequivalence vector spaces in the alternative set theory
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 1991
SP  - 517
EP  - 544
VL  - 32
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMUC_1991_32_3_a12/
LA  - en
ID  - CMUC_1991_32_3_a12
ER  - 
%0 Journal Article
%A Šmíd, Miroslav
%A Zlatoš, Pavol
%T Biequivalence vector spaces in the alternative set theory
%J Commentationes Mathematicae Universitatis Carolinae
%D 1991
%P 517-544
%V 32
%N 3
%U http://geodesic.mathdoc.fr/item/CMUC_1991_32_3_a12/
%G en
%F CMUC_1991_32_3_a12
Šmíd, Miroslav; Zlatoš, Pavol. Biequivalence vector spaces in the alternative set theory. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 3, pp. 517-544. http://geodesic.mathdoc.fr/item/CMUC_1991_32_3_a12/

[Dv 1977] Davis M.: Applied Nonstandard Analysis. Wiley Interscience, New York-London-Sydney. | MR | Zbl

[Dy 1940] Day M.M.: The spaces $L^p$ with $0< p< 1$. Bull. Amer. Math. Soc. 46 816-823. | MR

[En 1973] Enflo P.: A counterexample to the approximation problem in Banach spaces. Acta Math. 130 309-317. | MR | Zbl

[G-Z 1985a] Guričan J., Zlatoš P.: Biequivalences and topology in the alternative set theory. Comment. Math. Univ. Carolinae 26 525-552. | MR

[G-Z 1985b] Guričan J., Zlatoš P.: Archimedean and geodetical biequivalences. Comment. Math. Univ. Carolinae 26 675-698. | MR

[H-Mr 1972] Henson C.W., Moore L.C.: The nonstandard theory of topological vector spaces. Trans. Amer. Math. Soc. 172 405-435. | MR | Zbl

[H-Mr 1983a] Henson C.W., Moore L.C.: Nonstandard analysis and the theory of Banach spaces. in Hurd A.E. (ed.), ``Nonstandard Analysis - Recent Developments,'' Lecture Notes in Mathematics 983, pp. 27-112, Springer, Berlin-Heidelberg-New York-Tokyo. | MR | Zbl

[H-Mr 1983b] Henson C.W., Moore L.C.: The Banach spaces $\ell_p(n)$ for large $p$ and $n$. Manuscripta Math. 44 1-33. | MR

[K-Z 1988] Kalina M., Zlatoš P.: Arithmetic of cuts and cuts of classes. Comment. Math. Univ. Carolinae 29 435-456. | MR

[M 1979] Mlček J.: Valuations of structures. Comment. Math. Univ. Carolinae 20 525-552. | MR

[Rd 1982] Radyno Ya.V.: Linear Equations and Bornology (in Russian). Izdatelstvo Belgosuniversiteta, Minsk. | MR

[Rm 1980] Rampas Z.: Theory of matrices in the description of structures (in Czech). Master Thesis, Charles University, Prague.

[Rb-Rb 1964] Robertson A.P., Robertson W.J.: Topological Vector Spaces. Cambridge Univ. Press, Cambridge. | MR | Zbl

[Sn 1970] Singer I.: Bases in Banach Spaces I. Springer, Berlin-Heidelberg-New York. | MR | Zbl

[S-V 1980] Sochor A., Vopěnka P.: Revealments. Comment. Math. Univ. Carolinae 21 97-118. | MR

[V 1979] Vopěnka P.: Mathematics in the Alternative Set Theory. Teubner, Leipzig. | MR

[We 1976] Welsh D.J.A.: Matroid Theory. Academic Press, London-New York-San Francisco. | MR | Zbl

[Wi 1978] Wilansky A.: Modern Methods in Topological Vector Spaces. McGraw-Hill Int. Comp., New York-St.Louis. | MR | Zbl

[Z 1989] Zlatoš P.: Topological shapes. in Mlček J. et al. (eds.), ``Proc. of the $1^{st}$ Symposium on Mathematics in the Alternative Set Theory,'' pp. 95-120, Association of Slovak Mathematicians and Physicists, Bratislava.