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On 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 Phi:E^2 -> 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). 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.