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The closed graph theorem states that a linear operator between two Banach spaces X and Y is continuous iff it has a closed graph, where the "graph" {(x,f(x)):x in X} is ...

Cluster analysis is a technique used for classification of data in which data elements are partitioned into groups called clusters that represent collections of data elements ...

The compact-open topology is a common topology used on function spaces. Suppose X and Y are topological spaces and C(X,Y) is the set of continuous maps from f:X->Y. The ...

The companion matrix to a monic polynomial a(x)=a_0+a_1x+...+a_(n-1)x^(n-1)+x^n (1) is the n×n square matrix A=[0 0 ... 0 -a_0; 1 0 ... 0 -a_1; 0 1 ... 0 -a_2; | | ... ... |; ...

A completely multiplicative function, sometimes known as linear or totally multiplicative function, is an arithmetic function f(n) such that f(mn)=f(m)f(n) holds for each ...

A metric space X which is not complete has a Cauchy sequence which does not converge. The completion of X is obtained by adding the limits to the Cauchy sequences. For ...

Any computable function can be incorporated into a program using while-loops (i.e., "while something is true, do something else"). For-loops (which have a fixed iteration ...

A conical wedge is an ungula obtained by cutting a solid cone with a plane placed at an angle oblique to its base. For the special conical wedge obtained from a cone of ...

A topological space decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity, i.e., if a∼b and b∼c then a∼c. ...

A planar polygon is convex if it contains all the line segments connecting any pair of its points. Thus, for example, a regular pentagon is convex (left figure), while an ...