# Diagonal theorem

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) , .

The following diagonal theorem is a consequence of the preceding one ([a1], [a6], [a8]): Let be a commutative group with a quasi-norm , i.e.

and let be an infinite matrix in such that for every increasing sequence in there exists a subsequence of such that

Then .

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.

#### References

[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) |

**How to Cite This Entry:**

Diagonal theorem. E. Pap (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Diagonal_theorem&oldid=12262