Mazur-Orlicz theorem

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