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 $\mathcal{S}$ be a commutative semi-group with neutral element $0$ and with a triangular functional $f : \mathcal{S} \rightarrow [ 0 , + \infty )$, i.e.

$$\begin{equation*} f ( x ) - f ( y ) \leq f ( x + y ) \leq f ( x ) + f ( y ) , x , y \in \mathcal{S}, \end{equation*}$$

and $f ( 0 ) = 0$. For each sequence $\{ x_{j} \}$ in $\mathcal{S}$ and each $I \subset \mathbf{N}$, one writes $f ( \sum _ { j \in I } x _ { j } )$ for

$$\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \operatorname { sup } f \left( \sum _ { j \in I \bigcap [ 1 , n ] } x _ { j } \right) . \end{equation*}$$

The Mikusiński–Antosik–Pap diagonal theorem ([a1], [a4], [a5], [a6]) reads as follows. Let $( x _ { i j } )$ be an infinite matrix (indexed by ${\bf N} \times {\bf N}$) with entries in $\mathcal{S}$. Suppose that $\operatorname { lim } _ { n \rightarrow \infty } f ( x_{ij} ) = 0$, $i \in \mathbf{N}$. Then there exist an infinite set $I$ and a set $J \subset I$ such that

a) $\sum _ { j \in I } f ( x _ { i j } ) < \infty$, $i \in \mathbf{N}$; and

b) $f ( \sum _ { j \in J } x _ { i j } ) \geq f ( x _ { i i } ) / 2$, $i \in I$.

The following diagonal theorem is a consequence of the preceding one ([a1], [a6], [a8]): Let $( G , \| \, . \, \| )$ be a commutative group with a quasi-norm $\| \, . \, \| : G \rightarrow [ 0 , + \infty )$, i.e.

$$\begin{equation*} \| 0 \| = 0, \end{equation*}$$

$$\begin{equation*} \| - x \| = \| x \| , \| x + y \| \leq \| x \| + \| y \|, \end{equation*}$$

and let $( x _ { i j } )$ be an infinite matrix in $G$ such that for every increasing sequence $\{ m_i \}$ in $\mathbf{N}$ there exists a subsequence $\{ n _ { i } \}$ of $\{ m_i \}$ such that

$$\begin{equation*} \operatorname { lim } _ { i \rightarrow \infty } x _ { n _ { i } n _ { j }} = 0 \text { for all } j \in \mathbf{N}, \end{equation*}$$

$$\begin{equation*} \operatorname { lim } _ { i \rightarrow \infty } \sum _ { j = 1 } ^ { \infty } x _ { n_i n_j } = 0. \end{equation*}$$

Then $\operatorname { lim } _ { i \rightarrow \infty } x _ { i i } = 0$.

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)