Math 2.0

Sequences, series, summability General summability methods Theorem

Mazur–Orlicz theorem

Builds on Matrix summability methodRegular summability methodConsistency of summability methodsBounded sequence

Statement

Let A=(ank)A = (a_{nk}) and B=(bnk)B = (b_{nk}) be regular matrix summability methods. A sequence x=(xk)x = (x_k) is AA-summable if each series (Ax)n=∑kankxk(Ax)_n = \sum_{k} a_{nk} x_k converges and the limit lim⁡Ax=lim⁡n→∞(Ax)n\lim_A x = \lim_{n \to \infty} (Ax)_n exists; the same holds for BB. Assume that every bounded sequence that is AA-summable is also BB-summable. Then lim⁡Ax=lim⁡Bx\lim_A x = \lim_B x for every bounded AA-summable sequence xx.

Remarks

The result is also called the bounded consistency theorem: inclusion of the bounded summability fields forces consistency on them. Its proof is functional-analytic. It treats the bounded AA-summable sequences as a space of sequences with the supremum norm. On this space it studies the linear functional lim⁡B−lim⁡A\lim_B - \lim_A, which vanishes on convergent sequences because both methods are regular.

Example: every bounded sequence that is Cesàro summable of order 11 is Cesàro summable of order 22. The theorem then says at once that the two Cesàro limits agree on such sequences.

The conclusion is only about bounded sequences. The theorem says nothing about unbounded sequences in the summability field of AA.

Going deeper

For a student

The theorem belongs to the functional-analytic side of summability theory, where summability fields are studied as sequence spaces. It complements the Silverman–Toeplitz theorem, which characterizes regular matrices. Inclusion theorems between classical methods, such as Cesàro and Hölder methods, become consistency theorems on bounded sequences through it.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Consistency of summability methods, concept, no evidence levelConsistency of summab…conceptMatrix summability method, concept, no evidence levelMatrix summability me…conceptRegular summability method, concept, no evidence levelRegular summability m…conceptSequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionMazur–Orlicz theorem, statement, no evidence levelMazur–Orlicz theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Consistency of summability methods concept
  2. Matrix summability method concept
  3. Regular summability method concept
  4. Sequence definition
  5. Real sequence definition
  6. Bounded sequence definition
  7. Mazur–Orlicz theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Divergent_series
About this page
TagYE5439NH1T
URIm2://YE5439NH1T
Kindstatement, theorem
WritingW0 stub
MSC40D25, 40C05, 40H05
StatusThis copy of the site has no build data. Run m2 status YE5439NH1T.
AboutConsistency of summability methods
UsesMatrix summability method, Regular summability method, Consistency of summability methods, Bounded sequence
Changed2 November 2026

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