Geodesic
Biequivalence vector spaces in the alternative set theory
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 3, pp. 517-544.

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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 
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     title = {Biequivalence vector spaces in the alternative set theory},
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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/