GROSS SPACES

A Gross space is a vector space E of infinite dimension over some field F, which is endowed with a symmetric bilinear form φ : E² → F and has the property that every infinite dimensional subspace U ⊆ E satisfies dim U^⊥ < dim E. Gross spaces over uncountable fields exist (in certain dimensions) (see [G/O]). 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. This was shown in [B/G], [B/Sp] and [Sp2]. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals ω₁, a Gross space in dimension ω₁ 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.