Some Results on Generalized Bases for Linear Topological Spaces

Abstract

Having introduced the basis concept for Hilbert and Banach spaces, certain generalizations appear to be at least as important. First of all one discards of almost all requirements used for the definition of a basis. Beginning with the absolute minimum one takes a linear topological space X, does not use the concepts of totalness and countability and avoids all mention of series expansions. Thus, the starting point will be a family {x_λ} of elements of X and a biorthogonal family {f_λ} of continuous linear (coefficient) functionals on X. The biorthogonal system {x _λ ,f _λ } is said to be maximal if there is no biorthogonal system in which it is properly contained. A biorthogonal system with respect to X is called a generalized basis for X if, in addition, x ∊ X and f _λ (x) = 0 for all X implies x = 0. A generalized basis is always a maximal biorthogonal system. If the set of basis elements {x _λ } of a biorthogonal system {x _λ ,f _λ } in X is total in X, then {x _λ ,f _λ } is called a dual generalized basis for X. Moreover, if such a basis is also a generalized basis for X, it is called a Markushevich basis for X if the set {x _λ } is countable, and an extended Markushevich basis for X if {x _λ } is not countable. Finally, introducing again the concept of a series expansion for elements of X, then a Markushevich basis for X becomes a Schauder basis for X.