<P> This construction is used in numerous contexts . The Gelfand--Naimark--Segal construction is a particularly important example of the use of this technique . Another example is the representation of semi-definite kernels on arbitrary sets . </P> <P> Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero x there exists some y such that ⟨ x, y ⟩ ≠ 0, though y need not equal x; in other words, the induced map to the dual space V → V is injective . This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold . By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with nonzero weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index . Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above . Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions). </P> <P> Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism V → V) and thus hold more generally . </P> <P> The term "inner product" is opposed to outer product, which is a slightly more general opposite . Simply, in coordinates, the inner product is the product of a 1 × n covector with an n × 1 vector, yielding a 1 × 1 matrix (a scalar), while the outer product is the product of an m × 1 vector with a 1 × n covector, yielding an m × n matrix . Note that the outer product is defined for different dimensions, while the inner product requires the same dimension . If the dimensions are the same, then the inner product is the trace of the outer product (trace only being properly defined for square matrices). In a quip: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out". </P>

De ne an operation which takes the mean of a vector using the inner product