of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object. Vector spaces with additional structure From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures. A vector space may be given a partial order under which some vectors can be compared. For example, -dimensional real space can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions where denotes the positive part of and the negative part. Normed vector spaces and inner product spaces "Measuring" vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted and respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively. Coordinate space can be equipped with the standard dot product: In this reflects the common notion of the angle between two vectors and by the law of cosines: Because of this, two vectors satisfying are called orthogonal. An important variant of the standard dot product is used in Minkowski space: endowed with the Lorentz product In contrast to the standard dot product, it is not positive definite: also takes negative values, for example, for Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of special relativity. Topological vector spaces Convergence questions are treated by considering vector spaces carrying a compatible topology, a structure that allows one to talk about elements being close to each other. Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if and in and