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

DOI:
https://doi.org/10.1090/S0002-9947-96-01658-3

MathSciNet review:
1357403

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Abstract: A *Gross space* is a vector space of infinite dimension over some field , which is endowed with a symmetric bilinear form and has the property that every infinite dimensional subspace satisfies dim dim. Gross spaces over uncountable fields exist (in certain dimensions). 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 , a Gross space in dimension 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

Email:
shelah@math.huji.ac.il

**Otmar Spinas**

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

Email:
ospinas@math.uci.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01658-3

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