Difference between revisions of "Mazur-Orlicz theorem"
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| − | A sequence | + | {{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 | ||
| − | + | $$A(x)=\lim_i\sum_{k=1}^\infty a_{i,k}\xi_i.$$ | |
| − | Let | + | 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 | + | 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 | + | 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) | + | i) $\chi(A)\neq0$; |
| − | ii) | + | 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 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 | + | 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
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