Biequivalence vector spaces in the alternative set theory

Miroslav Šmíd; Pavol Zlatoš

Commentationes Mathematicae Universitatis Carolinae (1991)

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

Abstract

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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.

How to cite

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Šmíd, 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 -

References

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  15. Sochor A., Vopěnka P., Revealments, Comment. Math. Univ. Carolinae 21 97-118. MR0566243
  16. Vopěnka P., Mathematics in the Alternative Set Theory, Teubner, Leipzig. MR0581368
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