Difference between revisions of "Mazur-Orlicz theorem"
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Revision as of 18:53, 24 March 2012
A sequence is said to be summable to
by a method
given by an infinite matrix
, if
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Let be the set of all sequences summables by a method
. Such a method is said to be convergence preserving if
contains all convergent sequences (it is not assumed, however, that for a convergent sequence
one has
; if the latter holds,
is called a permanent summability method). For a convergence-preserving method
there is a well-defined quantity
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Let and
be convergence-preserving methods with
, and assume that for each convergent sequence
one has
. Then the Mazur–Orlicz theorem is usually given to the following statement: If every bounded sequence in
is in
, then also for these sequences
([a3], Thm. 2; see also [a1] and [a2]).
A related result is as follows. If is a convergence-preserving method, then
contains an unbounded sequence if either of the following is satisfied ([a3], Thm. 7):
i) ;
ii) and
contains a bounded divergent sequence. A permanent method
is said to be perfectly inconsistent if for each divergent sequence
in
there is a permanent method
with
and
.
A permanent method is perfectly inconsistent if and only if every sequence in
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 -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 |
Mazur-Orlicz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mazur-Orlicz_theorem&oldid=12108