Images of bilinear mappings into $\mathbf R^3$

It is well-known that the image of a multilinear mapping into a vector space need not be a subspace of its target space. It is, however, far from clear which subsets of the target space may be such images. For vector spaces over the real numbers we give a complete classification of the images of bilinear mappings into a three-dimensional vector space. In Theorem 2.8 we show that either the image of a bilinear mapping into a three-dimensional space is a subspace, or its complement is either the interior of a double elliptic cone, or a plane from which two lines intersecting at the origin have been removed. We also show (Theorem 2.2) that the image of any multilinear mapping into a two-dimensional space is necessarily a subspace. Our methods are elementary and free of tensor considerations.