Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

Tom 233 / 2016

S. Gabriyelyan, J. Kąkol, G. Plebanek Studia Mathematica 233 (2016), 119-139 MSC: Primary 46A04, 46B25, 54C35; Secondary 28C15. DOI: 10.4064/sm8289-4-2016 Opublikowany online: 19 May 2016

Streszczenie

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space $X$ is an Ascoli space if every compact subset $\mathcal {K}$ of $C_k(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{\mathbb {R}}$-space, hence any $k$-space, is Ascoli.

Let $X$ be a metrizable space. We prove that the space $C_{k}(X)$ is Ascoli iff $C_{k}(X)$ is a $k_{\mathbb {R}}$-space iff $X$ is locally compact. Moreover, $C_{k}(X)$ endowed with the weak topology is Ascoli iff $X$ is countable and discrete.

Using some basic concepts from probability theory and measure-theoretic properties of $\ell _1$, we show that the following assertions are equivalent for a Banach space $E$: (i) $E$ does not contain an isomorphic copy of $\ell _1$, (ii) every real-valued sequentially continuous map on the unit ball $B_{w}$ with the weak topology is continuous, (iii) $B_{w}$ is a $k_{\mathbb {R}}$-space, (iv) $B_{w}$ is an Ascoli space.

We also prove that a Fréchet lcs $F$ does not contain an isomorphic copy of $\ell _1$ iff each closed and convex bounded subset of $F$ is Ascoli in the weak topology. Moreover we show that a Banach space $E$ in the weak topology is Ascoli iff $E$ is finite-dimensional. We supplement the last result by showing that a Fréchet lcs $F$ which is a quojection is Ascoli in the weak topology iff $F$ is either finite-dimensional or isomorphic to $\mathbb {K}^{\mathbb {N}}$, where $\mathbb {K}\in \{\mathbb {R},\mathbb {C}\}$.

Autorzy

  • S. GabriyelyanDepartment of Mathematics
    Ben-Gurion University of the Negev
    Beer-Sheva, P.O. 653, Israel
    e-mail
  • J. KąkolFaculty of Mathematics and Informatics
    A. Mickiewicz University
    61-614 Poznań, Poland
    and
    Institute of Mathematics
    Czech Academy of Sciences
    Žitna 25, Praha 1, Czech Republic
    e-mail
  • G. PlebanekInstytut Matematyczny
    Uniwersytet Wrocławski
    50-384 Wrocław, Poland
    e-mail

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek