### On Gross Spaces

by Shelah and Spinas. [ShSi:468]

Transactions American Math Soc, 1996

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.

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