# Biequivalence vector spaces in the alternative set theory

Commentationes Mathematicae Universitatis Carolinae (1991)

- Volume: 32, Issue: 3, page 517-544
- ISSN: 0010-2628

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topŠmíd, Miroslav, and Zlatoš, Pavol. "Biequivalence vector spaces in the alternative set theory." Commentationes Mathematicae Universitatis Carolinae 32.3 (1991): 517-544. <http://eudml.org/doc/247319>.

@article{Šmíd1991,

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.},

author = {Šmíd, Miroslav, Zlatoš, Pavol},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {alternative set theory; biequivalence; vector space; monad; galaxy; symmetric Sd-closure; dual; valuation; norm; convex; basis; topological vector spaces; natural infinity; biequivalent vector spaces; alternative set theory; locally convex vector spaces; internal set theory; external sets; external classes; dual spaces; topological base},

language = {eng},

number = {3},

pages = {517-544},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

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

url = {http://eudml.org/doc/247319},

volume = {32},

year = {1991},

}

TY - JOUR

AU - Šmíd, Miroslav

AU - Zlatoš, Pavol

TI - Biequivalence vector spaces in the alternative set theory

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1991

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 32

IS - 3

SP - 517

EP - 544

AB - 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.

LA - eng

KW - alternative set theory; biequivalence; vector space; monad; galaxy; symmetric Sd-closure; dual; valuation; norm; convex; basis; topological vector spaces; natural infinity; biequivalent vector spaces; alternative set theory; locally convex vector spaces; internal set theory; external sets; external classes; dual spaces; topological base

UR - http://eudml.org/doc/247319

ER -

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