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.

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