ASYMPTOTIC MEANS OF BOUNDED SEQUENCES IN BANACH SPACES(NONLINEAR ANALYSIS AND CONVEX ANALYSIS)
Kazuo Hashimoto · Institutional Repositories DataBase (IRDB) · 1998
Always $X,$ $Y$ are Banach spaces and %, 7 are non-principal ultrafilters on $\mathrm{N}$ , the set of natural numbers.For a pair of a norm-bounded sequence $(x_{n})$ in $X$ and a non-principal ultrafilter %on $\mathrm{N}$ , denote $\tau_{X}(x)=\lim_{n,\%}|\}_{X_{n}}-x||$ for $x\in X$ .In other words, $\tau_{X}(x)=\int_{\mathrm{N}}||x_{n}-X||\lambda(dn)$ for $x\in X$ , where $\lambda$ is a purely finitely additive 0-1 measure on $2^{\mathrm{N}}$ defined by $\lambda(A)=1$ if $A\in\%,$ $\lambda(A)=0$ otherwise.Krivine and Maurey [5] called such a functional a type on $X$ .We here call $\tau_{X}$ an asymptotic mean of $(x_{n})$ along $\mathscr{U}$ on $X$ .Let $\mathrm{Y}$ be a closed linear subspace of a Banach space $X$ and $(x_{n})$ a bounded sequence in Y.We call the set $M$ ( $x_{n}$ , %, Y) $=\{a\in \mathrm{Y}$ : $\tau_{\mathrm{Y}}(y)\geq\tau_{Y}(a)$ for all $y\in Y$ } an asymptotic center of $(x_{n})$ along % with respect to Y.If $\mathrm{Y}$ is separable, then the set $M(x_{n}, \mathscr{U}, Y)$ coincides with the asymptotic center in the sense of Lim [6] of a subsequence $(x_{n_{k}})$ of $(x_{n})$ with respect to $Y$ .For a bounded sequence $(x_{n})$ in $X$ , we set $\omega(x_{n})=n=1\infty\cap\overline{\mathrm{C}\mathrm{o}}\{xk : k\geq n\}$ .For any relatively weakly compact sequence $(x_{n})$ in $X$ , w-lim $x_{n}$ denotes the weak-limit of $(x_{n})$ along a non-principal ultrafilter %on N. Similarly, n,%for any bounded sequence $(f_{n})$ in the dual space $X^{*},$ $w^{*}-\lim f_{n}$ denotes the weak*-limit of n,% $(f_{n})$ along a non-principal ultrafilter $\mathscr{U}$ on N.The duality mapping of a Banach space is a possibly multi-valued mapping $F_{X}$ from $X$ into its dual space $X^{*}$ which assigns to each $x\in X$ a subset of $X^{*}$ defined by $F_{X}(x)=\{f\in x*f:(x)=||x||^{2}=||f||^{2}\}$ .