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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 topology induced by a topological space X on a subset S. The open sets of S are the intersections S intersection U, where U is an open set of X. For example, in the ...

The term energy has an important physical meaning in physics and is an extremely useful concept. There are several forms energy defined in mathematics. In measure theory, let ...

In real and functional analysis, equicontinuity is a concept which extends the notion of uniform continuity from a single function to collection of functions. Given ...

Euclidean n-space is denoted R^n.

A Banach algebra is an algebra B over a field F endowed with a norm ||·|| such that B is a Banach space under the norm ||·|| and ||xy||<=||x||||y||. F is frequently taken to ...

Let T be an ergodic endomorphism of the probability space X and let f:X->R be a real-valued measurable function. Then for almost every x in X, we have 1/nsum_(j=1)^nf ...

A real vector bundle pi:E->M has an orientation if there exists a covering by trivializations U_i×R^k such that the transition functions are vector space ...

The topology on the Cartesian product X×Y of two topological spaces whose open sets are the unions of subsets A×B, where A and B are open subsets of X and Y, respectively. ...

A bilinear form on a real vector space is a function b:V×V->R that satisfies the following axioms for any scalar alpha and any choice of vectors v,w,v_1,v_2,w_1, and w_2. 1. ...