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The Euclidean metric is the function d:R^n×R^n->R that assigns to any two vectors in Euclidean n-space x=(x_1,...,x_n) and y=(y_1,...,y_n) the number ...

Baire's category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the ...

The study of the geometry of figures by algebraic representation and manipulation of equations describing their positions, configurations, and separations. Analytic geometry ...

A vector v on a Hilbert space H is said to be cyclic if there exists some bounded linear operator T on H so that the set of orbits {T^iv}_(i=0)^infty={v,Tv,T^2v,...} is dense ...

Given a contravariant basis {e^->_1,...,e^->_n}, its dual covariant basis is given by e^->^alpha·e^->_beta=g(e^->^alpha,e^->_beta)=delta_beta^alpha, where g is the metric and ...

A symbol used to represent the point and space groups (e.g., 2/m3^_). Some symbols have abbreviated form. The equivalence between Hermann-Mauguin symbols (a.k.a. ...

A bounded operator T:V->W between two Banach spaces satisfies the inequality ||Tv||<=C||v||, (1) where C is a constant independent of the choice of v in V. The inequality is ...

A set S is discrete in a larger topological space X if every point x in S has a neighborhood U such that S intersection U={x}. The points of S are then said to be isolated ...

The hemicube, which might also be called the square hemiprism, is a simple solid that serves as an example of one of the two topological classes of convex hexahedron having 7 ...

The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...