Associate space with respect to a semi-finite measureTalimdjioski, Filip (Avtor)
KandiÄ‡, Marko (Mentor)
semi-finite and localizable measuresBanach function spacesassociate spaceorder and $\sigma$-order continuityFatou propertyWe introduce the basic theory of vector lattices and order bounded operators. Order continuous operators, the order dual space, normed lattices and several variants of the Fatou property of normed lattices are studied. We then introduce normed function spaces, saturated function seminorms and the associate space to a normed function space, as well as semi-finite and localizable measures. We prove that the associate space $E'$ of an arbitrary saturated Banach function space $E$ with respect to a semi-finite measure $\mu$ equals the order continuous dual space $E_n^{^\sim}$ if and only if $E'$ has the strong Fatou property. If $E$ is furthermore $\sigma$-order continuous we prove that $E_n^{^\sim}$, the sigma-order continuous dual space $E_c^{^\sim}$ and the norm dual $E^*$ are equal which implies that in the previously mentioned result the equality $E' = E_n^{^\sim}$ can be replaced with $E' = E^*$. Also, if $\mu$ is localizable, we prove that $E'$ has the strong Fatou property which implies that $E' = E_n^{^\sim}$. Finally, we give an example where the aforementioned equality fails.20192019-09-12 07:46:34Magistrsko delo/naloga110110sl