View source for Mazur-Orlicz theorem

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

<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>

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

<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>

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]]]).

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):

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110100/m11010024.png" />;

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" />.

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).

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.

====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>