Dual system

In mathematics, a dual system, dual pair, or duality over a field is a triple consisting of two vector spaces and over and a non-degenerate bilinear map . Duality theory, the study of dual systems, is part of functional analysis. It is separate and distinct to Dual-system Theory in psychology. Pairings A or pair over a field is a triple which may also be denoted by consisting of two vector spaces and over (which this article assumes is the field either of real numbers or the complex numbers ). and a bilinear map , which is called the bilinear map associated with the pairing or simply the pairing's map/bilinear form. For every , define and for every define Every is a linear functional on and every is a linear functional on . Let where each of these sets forms a vector space of linear functionals. It is common practice to write instead of , in which case the pair is often denoted by rather than However, this article will reserve the use of for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject. Dual pairings A pairing is called a , a , or a over if the bilinear form is non-degenerate, which means that it satisfies the following two separation axioms: separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero , the map is not identically (i.e. there exists a such that ); separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero the map is not identically (i.e. there exists an such that ). In this case say that is non-degenerate, say that places and in duality (or in separated duality), and is called the duality pairing of the . Total subsets A subset of is called if for every , implies A total subset of is defined analogously (see footnote). Thus separates points of if and only if is a total subset of , and similarly for . Orthogonality The vectors and are called , written , if . Two subsets and are orthogonal, written , if ; that is, if for all and .