construction of Banach limit using limit along an ultrafilter
Construction of Banach limit using limit along an ultrafilter
The existence of Banach limit is proved in mathematical analysis
usually by Hahn-Banach theorem
. (This proof can be found e.g. in
[5], [2] or [4].) Here we
will show another approach using limit along a filter. In fact we
define it as an -limit of , where is the
Cesàro mean of the sequence
and is an arbitrary
ultrafilter on .
Theorem 1.
Let be a free ultrafilter on . Let be a bounded (http://planetmath.org/Bounded) real
sequence. Then the functional
is a Banach limit.
Proof.
We first observe that is defined. Let us denote . Since is bounded, the sequence is bounded as well. Every bounded sequence has a limit along any ultrafilter. This means, that exists.
To prove that is a Banach limit, we should verify its continuity, positivity, linearity, shift-invariance and to verify that it extends limits.
We first show the shift-invariance. By we denote the sequence
and we want to show . We observe that
. As the sequence is bounded, the last
expression converges to 0. Thus and .
The rest of the proof is relatively easy, we only need to use the basic properties of a limit along a filter and of Cesàro mean.
Continuity: .
Positivity and linearity follow from positivity and linearity of -limit.
Extends limit: If is a convergent sequence, then its
Cesàro mean is convergent to the same limit.
∎
References
- 1 B. Balcar and P. Štěpánek, Teorie množin, Academia, Praha, 1986 (Czech).
-
2
C. Costara and D. Popa, Exercises in functional analysis
, Kluwer, Dordrecht, 2003.
-
3
K. Hrbacek and T. Jech, Introduction to set theory
, Marcel Dekker, New York, 1999.
-
4
T. J. Morisson, Functional analysis: An introduction to
Banach space
theory, Wiley, 2000.
- 5 Ch. Swartz, An introduction to functional analysis, Marcel Dekker, New York, 1992.
Title | construction of Banach limit using limit along an ultrafilter |
---|---|
Canonical name | ConstructionOfBanachLimitUsingLimitAlongAnUltrafilter |
Date of creation | 2013-03-22 15:32:29 |
Last modified on | 2013-03-22 15:32:29 |
Owner | kompik (10588) |
Last modified by | kompik (10588) |
Numerical id | 8 |
Author | kompik (10588) |
Entry type | Application |
Classification | msc 03E99 |
Classification | msc 40A05 |
Related topic | BanachLimit |