Namespaces
Variants
Actions

Difference between revisions of "Diagonal theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 34 formulas out of 34 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
 
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 34 formulas, 34 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|done}}
 
A generic theorem generalizing the classical  "sliding hump"  method given by H. Lebesgue and O. Toeplitz, see [[#References|[a3]]], and very useful in the proof of generalized fundamental theorems of [[Functional analysis|functional analysis]] and [[Measure|measure]] theory.
 
A generic theorem generalizing the classical  "sliding hump"  method given by H. Lebesgue and O. Toeplitz, see [[#References|[a3]]], and very useful in the proof of generalized fundamental theorems of [[Functional analysis|functional analysis]] and [[Measure|measure]] theory.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201101.png" /> be a commutative [[Semi-group|semi-group]] with neutral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201102.png" /> and with a triangular functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201103.png" />, i.e.
+
Let $\mathcal{S}$ be a commutative [[Semi-group|semi-group]] with neutral element $0$ and with a triangular functional $f : \mathcal{S} \rightarrow [ 0 , + \infty )$, i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201104.png" /></td> </tr></table>
+
\begin{equation*} f ( x ) - f ( y ) \leq f ( x + y ) \leq f ( x ) + f ( y ) , x , y \in \mathcal{S}, \end{equation*}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201105.png" />. For each sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201106.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201107.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201108.png" />, one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d1201109.png" /> for
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011010.png" /></td> </tr></table>
+
\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 ([[#References|[a1]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]]) reads as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011011.png" /> be an infinite matrix (indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011012.png" />) with entries in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011013.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011015.png" />. Then there exist an infinite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011016.png" /> and a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011017.png" /> such that
+
The Mikusiński–Antosik–Pap diagonal theorem ([[#References|[a1]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011019.png" />; and
+
a) $\sum _ { j \in I } f ( x _ { i j } ) &lt; \infty$, $i \in \mathbf{N}$; and
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011021.png" />.
+
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 ([[#References|[a1]]], [[#References|[a6]]], [[#References|[a8]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011022.png" /> be a commutative [[Group|group]] with a [[Quasi-norm|quasi-norm]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011023.png" />, i.e.
+
The following diagonal theorem is a consequence of the preceding one ([[#References|[a1]]], [[#References|[a6]]], [[#References|[a8]]]): Let $( G , \| \, . \, \| )$ be a commutative [[Group|group]] with a [[Quasi-norm|quasi-norm]] $\| \, . \, \| : G \rightarrow [ 0 , + \infty )$, i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011024.png" /></td> </tr></table>
+
\begin{equation*} \| 0 \| = 0, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011025.png" /></td> </tr></table>
+
\begin{equation*} \| - x \| = \| x \| , \| x + y \| \leq \| x \| + \| y \|, \end{equation*}
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011026.png" /> be an infinite matrix in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011027.png" /> such that for every increasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011029.png" /> there exists a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011031.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011032.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { lim } _ { i \rightarrow \infty } x _ { n _ { i }  n _ { j }} = 0 \text { for all } j \in \mathbf{N}, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011033.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { lim } _ { i \rightarrow \infty } \sum _ { j = 1 } ^ { \infty } x _ { n_i n_j } = 0. \end{equation*}
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120110/d12011034.png" />.
+
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., [[#References|[a1]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a9]]]): the [[Nikodým convergence theorem|Nikodým convergence theorem]]; the [[Vitali–Hahn–Saks theorem|Vitali–Hahn–Saks theorem]]; the [[Nikodým boundedness theorem|Nikodým boundedness theorem]]; the uniform boundedness theorem (cf. [[Uniform boundedness|Uniform boundedness]]); the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]]; the Bourbaki theorem on joint continuity; the Orlicz–Pettis theorem (cf. [[Vector measure|Vector measure]]); the kernel theorem for sequence spaces; the Bessaga–Pelczynski theorem; the [[Pap adjoint theorem|Pap adjoint theorem]]; and the [[Closed-graph theorem|closed-graph theorem]].
 
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., [[#References|[a1]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a9]]]): the [[Nikodým convergence theorem|Nikodým convergence theorem]]; the [[Vitali–Hahn–Saks theorem|Vitali–Hahn–Saks theorem]]; the [[Nikodým boundedness theorem|Nikodým boundedness theorem]]; the uniform boundedness theorem (cf. [[Uniform boundedness|Uniform boundedness]]); the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]]; the Bourbaki theorem on joint continuity; the Orlicz–Pettis theorem (cf. [[Vector measure|Vector measure]]); the kernel theorem for sequence spaces; the Bessaga–Pelczynski theorem; the [[Pap adjoint theorem|Pap adjoint theorem]]; and the [[Closed-graph theorem|closed-graph theorem]].
Line 36: Line 44:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Antosik,  C. Swartz,  "Matrix methods in analysis" , ''Lecture Notes Math.'' , '''1113''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Diestel,  J.J. Uhl,  "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''I''' , Springer  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Pap,  "Functional analysis (Sequential convergence)" , Inst. Math. Novi Sad  (1982)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Pap,  C. Swartz,  "The closed graph theorem for locally convex spaces"  ''Boll. Un. Mat. Ital.'' , '''7''' :  4-B  (1990)  pp. 109–111</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.S. Sobolev,  "Introduction to cubature formulas" , Nauka  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  C. Swartz,  "Introduction to functional analysis" , M. Dekker  (1992)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Antosik,  C. Swartz,  "Matrix methods in analysis" , ''Lecture Notes Math.'' , '''1113''' , Springer  (1985)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Diestel,  J.J. Uhl,  "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  G. Köthe,  "Topological vector spaces" , '''I''' , Springer  (1969)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  E. Pap,  "Functional analysis (Sequential convergence)" , Inst. Math. Novi Sad  (1982)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  E. Pap,  C. Swartz,  "The closed graph theorem for locally convex spaces"  ''Boll. Un. Mat. Ital.'' , '''7''' :  4-B  (1990)  pp. 109–111</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  L.S. Sobolev,  "Introduction to cubature formulas" , Nauka  (1974)  (In Russian)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  C. Swartz,  "Introduction to functional analysis" , M. Dekker  (1992)</td></tr></table>

Latest revision as of 16:55, 1 July 2020

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)
How to Cite This Entry:
Diagonal theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_theorem&oldid=50100
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article