View source for Mazur-Orlicz theorem

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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 $A(x)=B(x)$ ([[#References|[a3]]], Thm. 2; see also [[#References|[a1]]] and [[#References|[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 ([[#References|[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 ([[#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 $F$-spaces (see [[Fréchet topology|Fréchet topology]]) is due to them.

====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>