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 ,  or .) 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 .
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.
Positivity and linearity follow from positivity and linearity of -limit.
- 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|
|Date of creation||2013-03-22 15:32:29|
|Last modified on||2013-03-22 15:32:29|
|Last modified by||kompik (10588)|