਀㰀栀琀洀氀㸀 ਀㰀琀椀琀氀攀㸀䄀琀氀愀猀㨀 䤀渀琀攀爀渀愀琀椀漀渀愀氀 䌀漀渀昀攀爀攀渀挀攀 漀渀 吀漀瀀漀氀漀最礀 愀渀搀 椀琀猀 䄀瀀瀀氀椀挀愀琀椀漀渀猀 2007 at Kyoto (Jointly with 4th Japan Mexico Topology Conference) -਀匀甀戀洀椀猀猀椀漀渀 䘀漀爀洀㰀⼀琀椀琀氀攀㸀 ਀㰀洀攀琀愀 栀琀琀瀀ⴀ攀焀甀椀瘀㴀∀䌀漀渀琀攀渀琀ⴀ吀礀瀀攀∀ 挀漀渀琀攀渀琀㴀∀琀攀砀琀⼀栀琀洀氀㬀 挀栀愀爀猀攀琀㴀椀猀漀ⴀ㠀㠀㔀㤀ⴀ㄀∀㸀 ਀㰀戀漀搀礀 戀最挀漀氀漀爀㴀∀⌀昀昀昀昀昀昀∀ 琀攀砀琀㴀∀⌀　　　　　　∀㸀㰀℀ⴀⴀ 挀漀洀洀攀渀琀攀搀 漀甀琀Ⰰ 昀攀樀Ⰰ ㄀　⼀㌀　⼀㈀　　㜀 Atlas home ||਀㰀愀 栀爀攀昀㴀∀栀琀琀瀀㨀⼀⼀愀琀氀愀猀ⴀ挀漀渀昀攀爀攀渀挀攀猀⸀挀漀洀⼀挀漀渀昀攀爀攀渀挀攀⸀栀琀洀∀㸀䌀漀渀昀攀爀攀渀挀攀猀㰀⼀愀㸀 簀 Abstracts |਀ about Atlas -->਀㰀栀爀 猀椀稀攀㴀㄀ 渀漀猀栀愀搀攀㸀 ਀㰀琀爀㸀㰀琀搀㸀㰀琀愀戀氀攀 戀漀爀搀攀爀㴀㄀ 挀攀氀氀瀀愀搀搀椀渀最㴀㐀 挀攀氀氀猀瀀愀挀椀渀最㴀　 眀椀搀琀栀㴀∀㄀　　─∀㸀 ਀㰀⼀琀愀戀氀攀㸀㰀⼀琀搀㸀

International Conference on Topology and its Applications਀㈀　　㜀 愀琀 䬀礀漀琀漀 ⠀䨀漀椀渀琀氀礀 眀椀琀栀 㐀琀栀 䨀愀瀀愀渀 䴀攀砀椀挀漀 吀漀瀀漀氀漀最礀 䌀漀渀昀攀爀攀渀挀攀⤀㰀⼀戀㸀㰀戀爀㸀 December 3-7, 2007
਀㰀戀㸀䐀攀瀀愀爀琀洀攀渀琀 漀昀 䴀愀琀栀攀洀愀琀椀挀猀Ⰰ 䬀礀漀琀漀 唀渀椀瘀攀爀猀椀琀礀㰀⼀戀㸀㰀戀爀㸀 Kyoto, Japan

Organizers
Chair: Akira Kono, Salvador Garcia-Ferreira;਀䄀氀最攀戀爀愀椀挀 吀漀瀀漀氀漀最礀㨀 一漀爀椀漀 䤀眀愀猀攀Ⰰ 䴀椀最甀攀氀 䄀⸀ 堀椀挀漀琀攀✀渀挀愀琀氀㬀 䬀渀漀琀 吀栀攀漀爀礀㨀 䄀欀椀漀 Kawauchi, Mario Eudave; Set Theory, Set-theoretic Topology: Tsugunori਀一漀最甀爀愀Ⰰ 䄀渀最攀氀 吀愀洀愀爀椀稀ⴀ䴀愀猀挀愀爀甀愀Ⰰ 䐀椀攀最漀 刀攀戀漀氀氀攀搀漀ⴀ刀漀樀愀猀㬀 䜀攀漀洀攀琀爀椀挀 吀漀瀀漀氀漀最礀Ⰰ Continuum Theory: Hisao Kato, Sergey Antonyan; Dynamical Systems: Hiroshi਀䬀漀欀甀戀甀㰀⼀戀㸀㰀⼀瀀㸀 ਀

Two converses to a refinement of the Hahn-Banach਀吀栀攀漀爀攀洀㰀⼀戀㸀㰀⼀戀椀最㸀㰀戀爀㸀 by
਀㰀戀椀最㸀㰀戀㸀䘀爀攀搀 䔀⸀䨀⸀ 䰀椀渀琀漀渀㰀⼀戀㸀㰀⼀戀椀最㸀㰀戀爀㸀圀攀猀氀攀礀愀渀 唀渀椀瘀攀爀猀椀琀礀Ⰰ 䴀椀搀搀氀攀琀漀眀渀Ⰰ 䌀吀Ⰰ USA [Emeritus]਀㰀⼀瀀㸀

A little-known mild refinement of the Hahn-Banach Theorem helps਀琀漀 挀栀愀爀愀挀琀攀爀椀稀攀 琀栀攀 爀攀愀氀 䈀愀渀愀挀栀 猀瀀愀挀攀猀 戀漀琀栀 昀爀漀洀 愀洀漀渀最猀琀 琀栀攀 爀攀愀氀 normed linear spaces, as well as from amongst the abstract algebras਀眀栀漀猀攀 漀瀀攀爀愀琀椀漀渀猀 愀爀攀 琀栀攀 渀愀琀甀爀愀氀 漀瀀攀爀愀琀椀漀渀猀Ⰰ 昀椀渀椀琀愀爀礀 愀渀搀 椀渀昀椀渀椀琀愀爀礀Ⰰ on Banach discs. These algebras, sometimes called convexoids,਀挀漀渀猀琀椀琀甀琀攀 琀栀攀 瘀愀爀椀攀琀愀氀 爀攀昀氀攀挀琀椀漀渀 漀昀 琀栀攀 挀愀琀攀最漀爀礀 漀昀 䈀愀渀愀挀栀 spaces; the natural operations referred to are the various਀㰀℀ⴀⴀ ∀ ⴀⴀ㸀☀氀搀焀甀漀㬀猀甀戀ⴀ挀漀渀瘀攀砀ⴀ挀漀洀戀椀渀愀琀椀漀渀㰀℀ⴀⴀ ∀ ⴀⴀ㸀☀爀搀焀甀漀㬀 漀瀀攀爀愀琀漀爀猀 愀爀椀猀椀渀最 昀爀漀洀 愀氀氀 琀栀攀 absolutely summable real sequences (finite or infinite)਀眀椀琀栀 ⠀氀㰀猀甀瀀㸀⠀㄀⤀㰀⼀猀甀瀀㸀⤀ 渀漀爀洀 ☀⌀㠀㠀　㐀㬀 ㄀⸀ ਀㰀瀀㸀 The Hahn-Banach Theorem itself, commonly read as asserting that਀琀栀攀 渀愀琀甀爀愀氀 㰀℀ⴀⴀ ∀ ⴀⴀ㸀☀氀搀焀甀漀㬀攀瘀愀氀甀愀琀椀漀渀 洀愀瀀㰀℀ⴀⴀ ∀ ⴀⴀ㸀☀爀搀焀甀漀㬀  iV: V → V** from਀愀渀礀 爀攀愀氀 䈀愀渀愀挀栀 猀瀀愀挀攀 嘀 琀漀 椀琀猀 猀攀挀漀渀搀 搀甀愀氀 嘀㰀猀甀瀀㸀⨀⨀㰀⼀猀甀瀀㸀 椀猀 愀渀 isometric embedding, is easily tweaked to reveal (and this is਀琀栀愀琀 洀椀氀搀 爀攀昀椀渀攀洀攀渀琀⤀ 琀栀愀琀 椀㰀猀甀戀㸀嘀㰀⼀猀甀戀㸀 椀猀 愀挀琀甀愀氀氀礀 愀渀 攀焀甀愀氀椀稀攀爀 (or “difference kernel”) of the ਀挀漀爀爀攀猀瀀漀渀搀椀渀最 攀瘀愀氀甀愀琀椀漀渀 洀愀瀀 i(V**): V** → V**** for਀嘀㰀猀甀瀀㸀⨀⨀㰀⼀猀甀瀀㸀 愀渀搀 琀栀攀 second transpose (iV)**: V** →਀嘀㰀猀甀瀀㸀⨀⨀⨀⨀㰀⼀猀甀瀀㸀 ਀漀昀 椀㰀猀甀戀㸀嘀㰀⼀猀甀戀㸀 椀琀猀攀氀昀⸀ ਀㰀瀀㸀 Fortunately, counterparts of these maps persist when V is merely਀愀 爀攀愀氀 渀漀爀洀攀搀 氀椀渀攀愀爀 猀瀀愀挀攀Ⰰ 愀渀搀 漀甀爀 昀椀爀猀琀 挀漀渀瘀攀爀猀攀 ⠀琀漀 琀栀攀 琀眀攀愀欀攀搀 Hahn-Banach Theorem) is then:਀

਀㰀攀洀㸀 吀栀攀 爀攀愀氀 渀漀爀洀攀搀 氀椀渀攀愀爀 猀瀀愀挀攀 嘀 椀猀 挀漀洀瀀氀攀琀攀㰀⼀攀洀㸀 ⠀㰀攀洀㸀椀⸀攀⸀Ⰰ 椀猀 愀氀爀攀愀搀礀 a real Banach space) if (and only਀椀昀⤀ 㰀攀洀㸀琀栀攀 洀愀瀀 椀㰀猀甀戀㸀嘀㰀⼀猀甀戀㸀 椀猀 愀渀 攀焀甀愀氀椀稀攀爀 漀昀 琀栀攀 pair ( i(V**) , (iV)** ).਀ ਀㰀瀀㸀 Counterparts of those same maps persist as well for convexoids V,਀愀渀搀 漀甀爀 猀攀挀漀渀搀 挀漀渀瘀攀爀猀攀 椀猀 琀栀攀渀㨀 ਀㰀瀀㸀 The convexoid V is (the unit disc਀漀昀㰀⼀攀洀㸀⤀ 㰀攀洀㸀愀 爀攀愀氀 䈀愀渀愀挀栀 猀瀀愀挀攀 椀昀㰀⼀攀洀㸀 ⠀愀渀搀 漀渀氀礀 if) the map iV is an equalizer of the਀瀀愀椀爀㰀⼀攀洀㸀 ⠀ 椀㰀猀甀戀㸀⠀嘀㰀猀甀瀀㸀⨀⨀㰀⼀猀甀瀀㸀⤀㰀⼀猀甀戀㸀 Ⰰ ⠀椀㰀猀甀戀㸀嘀㰀⼀猀甀戀㸀⤀㰀猀甀瀀㸀⨀⨀㰀⼀猀甀瀀㸀 ⤀⸀㰀⼀瀀㸀 ਀ ਀㰀瀀 愀氀椀最渀㴀爀椀最栀琀㸀㰀猀洀愀氀氀㸀䐀愀琀攀 爀攀挀攀椀瘀攀搀㨀 伀挀琀漀戀攀爀 ㈀㠀Ⰰ ㈀　　㜀㰀⼀猀洀愀氀氀㸀㰀⼀瀀㸀 ਀㰀栀爀 猀椀稀攀㴀㄀ 渀漀猀栀愀搀攀㸀㰀℀ⴀⴀ 挀漀洀洀攀渀琀攀搀 漀甀琀Ⰰ 昀攀樀Ⰰ ㄀　⼀㌀　⼀㈀　　㜀㨀 

You can now go back and edit your submission.਀伀爀Ⰰ 琀漀 挀漀洀瀀氀攀琀攀 礀漀甀爀 猀甀戀洀椀猀猀椀漀渀Ⰰ 礀漀甀 洀甀猀琀 挀漀渀琀椀渀甀攀 琀漀 琀栀攀 昀椀渀愀氀 瀀愀最攀⸀㰀戀爀㸀 Please note that our online previewing software may not show your TeX਀挀漀洀洀愀渀搀猀 properly, but your original abstract is always sent to the conference਀漀爀最愀渀椀稀攀爀猀⸀㰀⼀瀀㸀

਀㰀椀渀瀀甀琀 琀礀瀀攀㴀∀栀椀搀搀攀渀∀ 渀愀洀攀㴀∀猀琀愀琀攀∀ 瘀愀氀甀攀㴀∀㄀㄀㤀㌀㔀㔀　㔀　㠀∀㸀 ਀㰀⼀昀漀爀洀㸀 ਀㰀椀渀瀀甀琀 琀礀瀀攀㴀∀猀甀戀洀椀琀∀ 瘀愀氀甀攀㴀∀䌀漀渀琀椀渀甀攀 ⠀昀椀渀椀猀栀⤀ ☀最琀㬀☀最琀㬀∀㸀 ਀
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