Μελέτη γεωμετρικών ιδιοτήτων χώρων Banach

Abstract

Αφιερώνεται στον Μίνωα, την Αρετή και την Κλέα v Prìlogoc H paroÔsa Didaktorik Diatrib apoteleÐ thn katagraf twn apotelesmtwn thc ereunhtik c mou melèthc, gewmetrik¸n idiot twn q¸rwn Banach, upì thn kajod ghsh kai epÐbleyh tou Dasklou mou, EpÐkourou Kajhght sto PoluteqneÐo Kr thc, k. MÐnwa Petrkh. H pro-Perieqìmena EISAGWGH xiii Keflaio 1. STOIQEIA APO THN JEWRIA QWRWN BANACH 1.1. BasikoÐ orismoÐ, eujÔ ginìmeno, probolèc, • -topologÐa kai w-topologÐa, Je¸rhma Mazur 1.2. Bseic kai basikèc akoloujÐec, probolèc merik¸n ajroismtwn, isodunamÐa bsewn, block basikèc akoloujÐec, Principle of small pertrubations, arq thc epilog c twn Bessaga-Pelzynski, bseic se klasikoÔc q¸rouc, sÔsthma Haar 1.3. Unconditional, shrinking kai boundedly complete bseic, ènac qarakthrismìc gia touc autopajeÐc q¸rouc 1.4. DeÐktec, dèndra, d-bushes, d-approximate bushes, averaging back bushes 1.5. To arqètupo (fundamental) bush ston c 0 Keflaio 2. RADON-NIKODYM, KREIN-MILMAN KAI SUNAFEIS IDIOTHTES 2.1. Dianusmatik mètra, metr simec sunart seic, olokl rwma Bochner kai anaparastsimoi telestèc (RNP mèroc 1o) 2.2. Dianusmatik martingales kai dentability (RNP mèroc 2o) 2.3. Idiìthta Krein-Milman, h RNP sunepgetai thn KMR 2.4. H RNP eÐnai isodÔnamh me thn KMR stouc diaqwrÐsimouc duikoÔc (1975), sta Banach lattices (1981), ìtan o q¸roc eÐnai isìmorfoc me to tetrgwnì tou (1985) 2.5. H RNP eÐnai isodÔnamh me thn KMR sta strongly regular sÔnola (1985), sta uposÔnola q¸rwn me unconditional bsh (1985), point of continuity property 2.6. H RNP eÐnai isodÔnamh me thn KMR ìtan h asjen c kai h norm topologÐa sumpÐptoun (1989), sta non-dentable non-PCP uposÔnola q¸rwn me unconditional skipped block decomposition, denting points, an o q¸roc den èqei thn RNP tìte perièqei upìqwro me FDD pou den èqei thn RNP ix x PERIEQ OMENA 2.7. CFDSD, Pal representations, mia efarmog ston c 0 , h RNP eÐnai isodÔnamh me thn KMR sta uposÔnola tou jetikoÔ k¸nou tou L 1 (0, 1) (1993) Keflaio 3. KAJE NON-DENTABLE UPOSUNOLO TOU C(ω ω k ) PERIEQEI ENA KLEISTO KURTO UPOSUNOLO L TETOIO WSTE TO L EQEI THN PCP KAI OQI THN RNP. SUNEPWS H RNP KAI H KMR EINAI ISODUNAMES IDIOTHTES STA UPOSUNOLA TWN QWRWN BANACH C(ω ω k ) 3.1. Oi q¸roi Banach C(α) ìpou α arijm simoc diataktikìc arijmìc, h kathgoriopoÐhsh twn C(K) me K sumpag metrikì q¸ro (1960 kai 1966), mia sunèpeia ìtan o q¸roc Banach X den perièqei ton 1 (1978) 3.2. O Cantor-Bendixson deÐkthc, parathr seic, gnwst apotelèsmata gia K arijm simo sumpagèc 3.3. Mia anaparstash tou sunìlou 1,ω ω , o q¸roc C(ω ω ) den emfuteÔetai se q¸ro me unconditional bsh 3.4. Megloi telestèc ston L 1 (0, 1) me mikrèc probolèc, telestèc Dunford-Pettis 3.5. Block δ-approximate bushes, sunèpeiec thc unconditionality (ajroismtwn) se q¸rouc me Schauder dimensional decomposition 3.6. Sunèpeiec ìtan K kleistì, kurtì, fragmèno, non-PCP, uposÔnolo tou X kai o q¸roc Banach X den perièqei ton xvi EISAGWGH small combination of slices idiìthta. Tìte uprqei èna kleistì, kurtì, mh kenì uposÔnolo W tou K tètoio¸ste : ( * ) to W eÐnai non-dentable kai h norm me thn asjen topolo...