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Transactions of the American Mathematical Society

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Gross spaces

Authors: Saharon Shelah and Otmar Spinas
Journal: Trans. Amer. Math. Soc. 348 (1996), 4257-4277
MSC (1991): Primary 11E04, 03E35; Secondary 12L99, 15A36
MathSciNet review: 1357403
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Abstract: A Gross space is a vector space $E$ of infinite dimension over some field $F$, which is endowed with a symmetric bilinear form $\Phi :E^{2} \rightarrow F$ and has the property that every infinite dimensional subspace $U\subseteq E$ satisfies dim $U^{\perp }<$ dim $E$. Gross spaces over uncountable fields exist (in certain dimensions) (see [H. Gross and E. Ogg, Quadratic spaces with few isometries, Comment. Math. Helv. 48 (1973), 511-519]). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals $\omega _{1}$, a Gross space in dimension $\omega _{1}$ exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.

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Additional Information

Saharon Shelah
Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel
MR Author ID: 160185
ORCID: 0000-0003-0462-3152

Otmar Spinas
Affiliation: Department of Mathematics, University of California, Irvine, California 92717

Received by editor(s): August 1, 1995
Additional Notes: The authors are supported by the Basic Research Foundation of the Israel Academy of Science.
Article copyright: © Copyright 1996 American Mathematical Society