  A cubic space is a vector space equipped with a symmetric trilinear form. Two
cubic spaces are isogeneous if each embeds into the other. A cubic space is
non-degenerate if its form cannot be expressed as a finite sum of products of
linear and quadratic forms. We classify non-degenerate cubic spaces of
countable dimension up to isogeny: the isogeny classes are completely
determined by an invariant we call the residual rank, which takes values in
$\mathbf{N} \cup \{\infty\}$. In particular, the set of classes is discrete and
(under the partial order of embedability) satisfies the descending chain
condition.
