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 $\mathbb{F}$.... More 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. Less

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 $\mathbb{F}$.... More 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. Less

• 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