one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space and its dual, as above. Tensors in infinite dimensions This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are naturally isomorphic. Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to vector bundles or coherent sheaves. For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see topological tensor product). In some applications, it is the tensor product of Hilbert spaces that is intended, whose properties are the most similar to the finite-dimensional case. A more modern view is that it is the tensors' structure as a symmetric monoidal category that encodes their most important properties, rather than the specific models of those categories. Tensor fields In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space. This was the setting of Ricci's original work. In modern mathematical terminology such an object is called a tensor field, often referred to simply as a tensor. In this context, a coordinate basis is often chosen for the tangent vector space. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions, defining a coordinate transformation, Examples An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor because it changes its sign under those transformations that change the