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An operator T which commutes with all shift operators E^a, so TE^a=E^aT for all real a in a field. Any two shift-invariant operators commute.

The triangular inequalities are the inequalities |x-y|<=z<=x+y for real numbers (x,y,z) (Messiah 1962, p. 1056). If these inequalities hold for any one permutation of ...

An expression in mathematics which does not have meaning and so which is not assigned an interpretation. For example, division by zero is undefined in the field of real ...

A smooth structure on a topological manifold (also called a differentiable structure) is given by a smooth atlas of coordinate charts, i.e., the transition functions between ...

A complex manifold for which the exterior derivative of the fundamental form Omega associated with the given Hermitian metric vanishes, so dOmega=0. In other words, it is a ...

The complementary subspace problem asks, in general, which closed subspaces of a Banach space are complemented (Johnson and Lindenstrauss 2001). Phillips (1940) proved that ...

Let (X,tau) be a topological space, and let p in X. Then the arc component of p is union {A subset= X:A is an arc and p in A}.

Let f be a contraction mapping from a closed subset F of a Banach space E into F. Then there exists a unique z in F such that f(z)=z.

If X is any space, then there is a CW-complex Y and a map f:Y->X inducing isomorphisms on all homotopy, homology, and cohomology groups.

The space of continuously differentiable functions is denoted C^1, and corresponds to the k=1 case of a C-k function.