Izometrično injektivni Banachovi prostori

master's thesis

• Author: Pero Skoko
• Mentor: Ilja Gogić (mentor)
• Committee member: Ilja Gogić (predsjednik povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu proučavamo izometrično injektivne Banachove prostore, koji su definirani kao Banachovi prostori \(X\) nad poljem \(\mathbb{F} \in \{\mathbb{R}, \mathbb{C} \}\) takvi da za sve Banachove prostore \(Y\), zatvorene potprostore \(E \subseteq Y\) i ograničene operatore \(T : E \to X\), postoji ograničeno linearno proširenje \(\tilde{T} : Y \to X\) od \(T\) takvo da je \(\|\tilde{T}\| = \|T\|\). Prema Hahn-Banachovom teoremu osnovni primjer takvog prostora je samo polje \(\mathbb{F}\). Dokazat ćemo Goodner-Nachbin-Kellyjev teorem koji daje njihovu karakterizaciju: \(X\) je izometrično injektivan ako i samo ako je izometrično izomorfan prostoru \(\mathcal{C}(K)\) neprekidnih funkcija \(f: K \to \mathbb{F}\), pri čemu je \(K\) ekstremno nepovezan kompaktan Hausdorffov prostor. Navest ćemo neke netrivijalne primjere izometrično injektivnih Banachovih prostora. Kao interesantnu posljedicu dobivenih rezultata i Pełczynskijevove dekompozicijske tehnike, dokazat ćemo da su prostori \(\ell_\infty\) i \(L_\infty[0, 1]\) međusobno izomorfni.

Abstract (english)

In this thesis, we study isometrically injective Banach spaces, which are defined as Banach spaces \(X\) over the field \(\mathbb{F} \in \{\mathbb{R}, \mathbb{C} \}\) such that for all Banach spaces \(Y\), closed subspaces \(E \subseteq Y\), and bounded linear maps \(T : E \to X\), there exists a bounded linear extension \(\tilde{T} : Y \to X\) of \(T\) such that \(\|\tilde{T}\| = \|T\|\). By the Hahn-Banach theorem, the basic example of such a space is the underlying field \(\mathbb{F}\). We will prove the Goodner-Nachbin-Kelly theorem, which provides their characterization: \(X\) is isometrically injective if and only if it is isometrically isomorphic to a space \(\mathcal{C}(K)\) of continuous functions \(f: K \to \mathbb{F}\), where \(K\) is an extremally disconnected compact Hausdorff space. We will also provide some non-trivial examples of isometrically injective Banach spaces. As an interesting consequence of the obtained results and Pełczynski's decomposition technique, we will demonstrate that the spaces \(\ell_\infty\) i \(L_\infty[0, 1]\) are mutually isomorphic.

• Keywords: Banachovi prostori; Goodner-Nachbin-Kellyjev teorem
• Keywords (english): Banach spaces; Goodner-Nachbin-Kelly theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:906051
• Study programme: Title: Theoretical Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-08 10:40:29