Timeline for Cesaro means and Banach limits

10 events

when toggle format	what		by	license	comment
Jan 11, 2013 at 1:03	answer	added	Daniel Mansfield		timeline score: 4
Apr 9, 2012 at 13:47	comment	added	Martin Sleziak		Sequences with this property are called almost convergent and there exists a well-known characterization of such sequences due to Lorenz; which is described in the Wikipedia article. One possibility how to show this is using Hahn-Banach theorem, see this answer at Math.SE. Also the references from the Wikipedia article might be useful.
Apr 9, 2012 at 10:44	vote	accept	kap44
Apr 9, 2012 at 6:10	answer	added	Aaron Tikuisis		timeline score: 13
Apr 8, 2012 at 21:21	comment	added	Liviu Nicolaescu		Yes, thanks. $L$ is therefore an element of the topological dual of $\ell^\infty$.
Apr 8, 2012 at 20:40	history	edited	Yemon Choi		added FA tag
Apr 8, 2012 at 20:34	comment	added	kap44		If $L$ is a Banach limit, then $\|Lx\|\le\\|x\\|_\infty$ for any sequence $x$. Does this answer your question?
Apr 8, 2012 at 20:29	comment	added	Liviu Nicolaescu		I like the question, but I would like to make sure I understand correctly the notion of Banach limit. From your formulation I understand that a Banach limit is a linear map $L:\ell^infty(\mathbb{R})\to\mathbb{R}$ with the property that $L(\;(x_n)_{n\geq 0}\;)=L(\;(y_n)_{n\geq 0}\;) $ if $x_n=y_n$ for all $n$ sufficiently large. Is the functional $L$ continuous in the $\ell^\infty$ topology?
Apr 8, 2012 at 20:10	history	edited	kap44	CC BY-SA 3.0	added 24 characters in body
Apr 8, 2012 at 19:58	history	asked	kap44	CC BY-SA 3.0