Abstract: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.
Keywords: alternative set theory, biequivalence, vector space, monad, galaxy, symmetric Sd-closure, dual, valuation, norm, convex, basis
AMS Subject Classification: Primary 46Q05, 46A06, 46A35; Secondary 03E70, 03H05, 46A09