# Article

Full entry | PDF   (0.3 MB)
Keywords:
alternative set theory; biequivalence; vector space; monad; galaxy; symmetric Sd-closure; dual; valuation; norm; convex; basis
Summary:
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.
References:
[Dv 1977] Davis M.: Applied Nonstandard Analysis. Wiley Interscience, New York-London-Sydney. MR 0505473 | Zbl 0359.02060
[Dy 1940] Day M.M.: The spaces $L^p$ with $0< p< 1$. Bull. Amer. Math. Soc. 46 816-823. MR 0002700
[En 1973] Enflo P.: A counterexample to the approximation problem in Banach spaces. Acta Math. 130 309-317. MR 0402468 | Zbl 0286.46021
[G-Z 1985a] Guričan J., Zlatoš P.: Biequivalences and topology in the alternative set theory. Comment. Math. Univ. Carolinae 26 525-552. MR 0817825
[G-Z 1985b] Guričan J., Zlatoš P.: Archimedean and geodetical biequivalences. Comment. Math. Univ. Carolinae 26 675-698. MR 0831804
[H-Mr 1972] Henson C.W., Moore L.C.: The nonstandard theory of topological vector spaces. Trans. Amer. Math. Soc. 172 405-435. MR 0308722 | Zbl 0274.46013
[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 0698954 | Zbl 0511.46070
[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 0709841
[K-Z 1988] Kalina M., Zlatoš P.: Arithmetic of cuts and cuts of classes. Comment. Math. Univ. Carolinae 29 435-456. MR 0972828
[M 1979] Mlček J.: Valuations of structures. Comment. Math. Univ. Carolinae 20 525-552. MR 0555183
[Rd 1982] Radyno Ya.V.: Linear Equations and Bornology (in Russian). Izdatelstvo Belgosuniversiteta, Minsk. MR 0685429
[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 0162118 | Zbl 0423.46001
[Sn 1970] Singer I.: Bases in Banach Spaces I. Springer, Berlin-Heidelberg-New York. MR 0298399 | Zbl 0198.16601
[S-V 1980] Sochor A., Vopěnka P.: Revealments. Comment. Math. Univ. Carolinae 21 97-118. MR 0566243
[V 1979] Vopěnka P.: Mathematics in the Alternative Set Theory. Teubner, Leipzig. MR 0581368
[We 1976] Welsh D.J.A.: Matroid Theory. Academic Press, London-New York-San Francisco. MR 0427112 | Zbl 0343.05002
[Wi 1978] Wilansky A.: Modern Methods in Topological Vector Spaces. McGraw-Hill Int. Comp., New York-St.Louis. MR 0518316 | Zbl 0395.46001
[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.

Partner of