## Gross spaces

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- by Saharon Shelah and Otmar Spinas PDF
- Trans. Amer. Math. Soc.
**348**(1996), 4257-4277 Request permission

## 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
- Email: shelah@math.huji.ac.il
**Otmar Spinas**- Affiliation: Department of Mathematics, University of California, Irvine, California 92717
- Email: ospinas@math.uci.edu
- Received by editor(s): August 1, 1995
- Additional Notes: The authors are supported by the Basic Research Foundation of the Israel Academy of Science.
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**348**(1996), 4257-4277 - MSC (1991): Primary 11E04, 03E35; Secondary 12L99, 15A36
- DOI: https://doi.org/10.1090/S0002-9947-96-01658-3
- MathSciNet review: 1357403