Namespaces
Variants
Actions

Difference between revisions of "Mazur-Orlicz theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
 
Line 1: Line 1:
A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101001.png" /> is said to be summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101002.png" /> by a method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101003.png" /> given by an infinite matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101004.png" />, if
+
{{TEX|done}}
 +
A sequence $x=(\xi_i)_1^\infty$ is said to be summable to $A(x)$ by a method $A$ given by an infinite matrix $(a_{i,k})$, if
  
<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/m/m110/m110100/m1101005.png" /></td> </tr></table>
+
$$A(x)=\lim_i\sum_{k=1}^\infty a_{i,k}\xi_i.$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101006.png" /> be the set of all sequences summables by a method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101007.png" />. Such a method is said to be convergence preserving if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101008.png" /> contains all convergent sequences (it is not assumed, however, that for a convergent sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m1101009.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010010.png" />; if the latter holds, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010011.png" /> is called a permanent summability method). For a convergence-preserving method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010012.png" /> there is a well-defined quantity
+
Let $A^*$ be the set of all sequences summables by a method $A$. Such a method is said to be convergence preserving if $A^*$ contains all convergent sequences (it is not assumed, however, that for a convergent sequence $x=(\xi_i)$ one has $A(x)=\lim\xi_i$; if the latter holds, $A$ is called a permanent summability method). For a convergence-preserving method $A$ there is a well-defined quantity
  
<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/m/m110/m110100/m11010013.png" /></td> </tr></table>
+
$$\chi(A)=\lim_i\sum_ka_{i,k}-\sum_ka_{i,k}.$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010015.png" /> be convergence-preserving methods with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010016.png" />, and assume that for each convergent sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010017.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010018.png" />. Then the Mazur–Orlicz theorem is usually given to the following statement: If every bounded sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010019.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010020.png" />, then also for these sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010021.png" /> ([[#References|[a3]]], Thm. 2; see also [[#References|[a1]]] and [[#References|[a2]]]).
+
Let $A$ and $B$ be convergence-preserving methods with $\chi(A)\neq0$, and assume that for each convergent sequence $x$ one has $A(x)=B(x)$. Then the Mazur–Orlicz theorem is usually given to the following statement: If every bounded sequence in $A^*$ is in $B^*$, then also for these sequences $A(x)=B(x)$ ([[#References|[a3]]], Thm. 2; see also [[#References|[a1]]] and [[#References|[a2]]]).
  
A related result is as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010022.png" /> is a convergence-preserving method, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010023.png" /> contains an unbounded sequence if either of the following is satisfied ([[#References|[a3]]], Thm. 7):
+
A related result is as follows. If $A$ is a convergence-preserving method, then $A^*$ contains an unbounded sequence if either of the following is satisfied ([[#References|[a3]]], Thm. 7):
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010024.png" />;
+
i) $\chi(A)\neq0$;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010026.png" /> contains a bounded divergent sequence. A permanent method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010027.png" /> is said to be perfectly inconsistent if for each divergent sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010029.png" /> there is a permanent method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010030.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010032.png" />.
+
ii) $\chi(A)=0$ and $A^*$ contains a bounded divergent sequence. A permanent method $A$ is said to be perfectly inconsistent if for each divergent sequence $x$ in $A^*$ there is a permanent method $B$ with $B^*\supset A^*$ and $A(x)\neq B(x)$.
  
A permanent method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010033.png" /> is perfectly inconsistent if and only if every sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010034.png" /> is either convergent or unbounded ([[#References|[a3]]], Thm. 10).
+
A permanent method $A$ is perfectly inconsistent if and only if every sequence in $A^*$ is either convergent or unbounded ([[#References|[a3]]], Thm. 10).
  
S. Mazur and W. Orlicz also worked also in [[Functional analysis|functional analysis]]; e.g., the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010035.png" />-spaces (see [[Fréchet topology|Fréchet topology]]) is due to them.
+
S. Mazur and W. Orlicz also worked also in [[Functional analysis|functional analysis]]; e.g., the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] for $F$-spaces (see [[Fréchet topology|Fréchet topology]]) is due to them.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.L. Brudno,  "Summability of bounded sequences by means of matrices"  ''Mat. Sb.'' , '''16'''  (1949)  pp. 191–247  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Mazur,  W. Orlicz,  "Sur les mèthodes linèaires de sommation"  ''C.R. Acad. Sci. Paris'' , '''196'''  (1933)  pp. 32–34</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Mazur,  W. Orlicz,  "On linear methods of summability"  ''Studia Math.'' , '''14'''  (1954)  pp. 129–160</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.L. Brudno,  "Summability of bounded sequences by means of matrices"  ''Mat. Sb.'' , '''16'''  (1949)  pp. 191–247  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Mazur,  W. Orlicz,  "Sur les mèthodes linèaires de sommation"  ''C.R. Acad. Sci. Paris'' , '''196'''  (1933)  pp. 32–34</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Mazur,  W. Orlicz,  "On linear methods of summability"  ''Studia Math.'' , '''14'''  (1954)  pp. 129–160</TD></TR></table>

Latest revision as of 14:16, 3 August 2014

A sequence $x=(\xi_i)_1^\infty$ is said to be summable to $A(x)$ by a method $A$ given by an infinite matrix $(a_{i,k})$, if

$$A(x)=\lim_i\sum_{k=1}^\infty a_{i,k}\xi_i.$$

Let $A^*$ be the set of all sequences summables by a method $A$. Such a method is said to be convergence preserving if $A^*$ contains all convergent sequences (it is not assumed, however, that for a convergent sequence $x=(\xi_i)$ one has $A(x)=\lim\xi_i$; if the latter holds, $A$ is called a permanent summability method). For a convergence-preserving method $A$ there is a well-defined quantity

$$\chi(A)=\lim_i\sum_ka_{i,k}-\sum_ka_{i,k}.$$

Let $A$ and $B$ be convergence-preserving methods with $\chi(A)\neq0$, and assume that for each convergent sequence $x$ one has $A(x)=B(x)$. Then the Mazur–Orlicz theorem is usually given to the following statement: If every bounded sequence in $A^*$ is in $B^*$, then also for these sequences $A(x)=B(x)$ ([a3], Thm. 2; see also [a1] and [a2]).

A related result is as follows. If $A$ is a convergence-preserving method, then $A^*$ contains an unbounded sequence if either of the following is satisfied ([a3], Thm. 7):

i) $\chi(A)\neq0$;

ii) $\chi(A)=0$ and $A^*$ contains a bounded divergent sequence. A permanent method $A$ is said to be perfectly inconsistent if for each divergent sequence $x$ in $A^*$ there is a permanent method $B$ with $B^*\supset A^*$ and $A(x)\neq B(x)$.

A permanent method $A$ is perfectly inconsistent if and only if every sequence in $A^*$ is either convergent or unbounded ([a3], Thm. 10).

S. Mazur and W. Orlicz also worked also in functional analysis; e.g., the Banach–Steinhaus theorem for $F$-spaces (see Fréchet topology) is due to them.

References

[a1] A.L. Brudno, "Summability of bounded sequences by means of matrices" Mat. Sb. , 16 (1949) pp. 191–247 (In Russian)
[a2] S. Mazur, W. Orlicz, "Sur les mèthodes linèaires de sommation" C.R. Acad. Sci. Paris , 196 (1933) pp. 32–34
[a3] S. Mazur, W. Orlicz, "On linear methods of summability" Studia Math. , 14 (1954) pp. 129–160
How to Cite This Entry:
Mazur-Orlicz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mazur-Orlicz_theorem&oldid=22801
This article was adapted from an original article by W. Zelazko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article