A generic theorem generalizing the classical "sliding hump" method given by H. Lebesgue and O. Toeplitz, see [a3], and very useful in the proof of generalized fundamental theorems of functional analysis and measure theory.
Let be a commutative semi-group with neutral element and with a triangular functional , i.e.
and . For each sequence in and each , one writes for
The Mikusiński–Antosik–Pap diagonal theorem ([a1], [a4], [a5], [a6]) reads as follows. Let be an infinite matrix (indexed by ) with entries in . Suppose that , . Then there exist an infinite set and a set such that
a) , ; and
b) , .
and let be an infinite matrix in such that for every increasing sequence in there exists a subsequence of such that
Proofs involving diagonal theorems are characterized not only by simplicity but also by the possibility of further generalization. A great number of fundamental theorems in functional analysis and measure theory have been proven by means of diagonal theorems, such as (see, e.g., [a1], [a5], [a6], [a7], [a9]): the Nikodým convergence theorem; the Vitali–Hahn–Saks theorem; the Nikodým boundedness theorem; the uniform boundedness theorem (cf. Uniform boundedness); the Banach–Steinhaus theorem; the Bourbaki theorem on joint continuity; the Orlicz–Pettis theorem (cf. Vector measure); the kernel theorem for sequence spaces; the Bessaga–Pelczynski theorem; the Pap adjoint theorem; and the closed-graph theorem.
Rosenthal's lemma [a2] is closely related to diagonal theorems. Many related results can be found in [a1], [a2] [a5], [a6], [a9], where the method of diagonal theorems is used instead of the usually used Baire category theorem, which is equivalent with a weaker form of the axiom of choice.
See also Brooks–Jewett theorem.
|[a1]||P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)|
|[a2]||J. Diestel, J.J. Uhl, "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977)|
|[a3]||G. Köthe, "Topological vector spaces" , I , Springer (1969)|
|[a4]||J. Mikusiński, "A theorem on vector matrices and its applications in measure theory and functional analysis" Bull. Acad. Polon. Sci. Ser. Math. , 18 (1970) pp. 193–196|
|[a5]||E. Pap, "Functional analysis (Sequential convergence)" , Inst. Math. Novi Sad (1982)|
|[a6]||E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995)|
|[a7]||E. Pap, C. Swartz, "The closed graph theorem for locally convex spaces" Boll. Un. Mat. Ital. , 7 : 4-B (1990) pp. 109–111|
|[a8]||L.S. Sobolev, "Introduction to cubature formulas" , Nauka (1974) (In Russian)|
|[a9]||C. Swartz, "Introduction to functional analysis" , M. Dekker (1992)|
Diagonal theorem. E. Pap (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Diagonal_theorem&oldid=12262