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The coordinates representing any point of an n-dimensional affine space A by an n-tuple of real numbers, thus establishing a one-to-one correspondence between A and R^n. If V ...

A vector space V with a ring structure and a vector norm such that for all v,W in V, ||vw||<=||v||||w||. If V has a multiplicative identity 1, it is also required that ...

A linear functional on a real vector space V is a function T:V->R, which satisfies the following properties. 1. T(v+w)=T(v)+T(w), and 2. T(alphav)=alphaT(v). When V is a ...

Let Omega be a space with measure mu>=0, and let Phi(P,Q) be a real function on the product space Omega×Omega. When (mu,nu) = intintPhi(P,Q)dmu(Q)dnu(P) (1) = ...

A "curve" (i.e., a continuous map of a one-dimensional interval) into a two-dimensional area (a plane-filling function) or a three-dimensional volume.

For any sequence of integers 0<n_1<...<n_k, there is a flag manifold of type (n_1, ..., n_k) which is the collection of ordered sets of vector subspaces of R^(n_k) (V_1, ..., ...

A letter of the alphabet drawn with doubled vertical strokes is called doublestruck, or sometimes blackboard bold (because doublestruck characters provide a means of ...

A functional is a real-valued function on a vector space V, usually of functions. For example, the energy functional on the unit disk D assigns a number to any differentiable ...

A metric topology induced by the Euclidean metric. In the Euclidean topology of the n-dimensional space R^n, the open sets are the unions of n-balls. On the real line this ...

The standard Gauss measure of a finite dimensional real Hilbert space H with norm ||·||_H has the Borel measure mu_H(dh)=(sqrt(2pi))^(-dim(H))exp(1/2||h||_H^2)lambda_H(dh), ...