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 spa
ce. 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 ei
ther the interior of a double elliptic cone, or a plane from which two
lines intersecting at the origin have been removed. We also show (The
orem 2.2) that the image of any multilinear mapping into a two-dimensi
onal space is necessarily a subspace. Our methods are elementary and f
ree of tensor considerations.