International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 26

The set can be endowed with the structure of a vector space of dimension n over , and can be made into a Euclidean space by treating its standard basis as an orthonormal basis and defining the Euclidean norm:
By misuse of notation, x will sometimes also designate the column vector of coordinates of ; if these coordinates are referred to an orthonormal basis of , then where denotes the transpose of x.
The distance between two points x and y defined by allows the topological structure of to be transferred to , making it a metric space. The basic notions in a metric space are those of neighbourhoods, of open and closed sets, of limit, of continuity, and of convergence (see Section 1.3.2.2.6.1).
A subset S of is bounded if sup as x and y run through S; it is closed if it contains the limits of all convergent sequences with elements in S. A subset K of which is both bounded and closed has the property of being compact, i.e. that whenever K has been covered by a family of open sets, a finite subfamily can be found which suffices to cover K. Compactness is a very useful topological property for the purpose of proof, since it allows one to reduce the task of examining infinitely many local situations to that of examining only finitely many of them.