Topological Methods in Nonlinear Analysis

Function bases for topological vector spaces

Yilmaz Yilmaz

Full-text: Open access

Abstract

Our main interest in this work is to characterize certain operator spaces acting on some important vector-valued function spaces such as $(V_{a})_{c_{0}}^{a\in{\mathbb A}}$, by introducing a new kind basis notion for general Topological vector spaces. Where ${\mathbb A}$ is an infinite set, each $V_{a}$ is a Banach space and $(V_{a})_{c_{0}}^{a\in{\mathbb A}}$ is the linear space of all functions $x\colon{\mathbb A}\rightarrow\bigcup V_{a}$ such that, for each $\varepsilon> 0$, the set $\{a\in{\mathbb A}:\| x_{a}\| > \varepsilon\}$ is finite or empty. This is especially important for the vector-valued sequence spaces $(V_{i})_{c_{0}}^{i\in{\mathbb{N}}}$ because of its fundamental place in the theory of the operator spaces (see, for example, [H. P. Rosenthal, The complete separable extension property, J. Oper. Theory, 43 (2000), no. 2, 329-374]).

Article information

Source
Topol. Methods Nonlinear Anal., Volume 33, Number 2 (2009), 335-353.

Dates
First available in Project Euclid: 27 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461785642

Mathematical Reviews number (MathSciNet)
MR2549623

Zentralblatt MATH identifier
1188.46004

Citation

Yilmaz, Yilmaz. Function bases for topological vector spaces. Topol. Methods Nonlinear Anal. 33 (2009), no. 2, 335--353. https://projecteuclid.org/euclid.tmna/1461785642


Export citation

References

  • N. Bourbaki, General Topology, Reading Mass., Addison–Wesley (1966) \ref\key 2
  • L. Drewnowski, et al., On the Barelledness of spaces of bounded vector functions, Arch. Math., 63 (1994), 449–458 \ref\key 3
  • J. C. Ferrando, On the barelledness of the vector-valued bounded function space, J. Math. Anal. Appl., 184 (1994), 437–440 \ref\key 4 ––––, Unordered Baire-like vector-valued function spaces , Bull. Belg. Math. Soc., 2 (1995), 223–227 \ref\key 5 ––––, On some spaces of the vector-valued bounded functions, Math. Scand., 84 (1999), 71–80 \ref\key 6
  • J. C. Ferrando and S. V. Ludkowsky, Some Barelledness properties of $c_{0}( \Omega,X)$ , J. Math. Anal. Appl. 274(2002), 577–585 \ref\key 7
  • J. C. Ferrando and L. M. Sánchez Ruiz, Barrelledness in $\ell_{\infty}(\Omega,X) $ subspaces, J. Math. Anal. Appl., 303 (2005), 486–491 \ref\key 8
  • J. L. Kelley, General Topology, Princeton, N. J., Van Nostrand (1955) \ref\key 9
  • J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces \rom1, Springer–Verlag, Berlin, Heidelberg, New York (1977) \ref\key 10
  • I. J. Maddox, Elements of Functional Analysis, second ed., Cambridge Univ. Press, Cambridge, New York (1988) \ref\key 11
  • J. Mendoza, A barelledness criterion for $C_{0}(E) $ , Arch. Math., 40 (1983), 156–158 \ref\key 12
  • H. P. Rosenthal, The complete separable extension property, J. Oper. Theory, 43 , no. 2, 329–374 (2000) \ref\key 13
  • I. Singer, Bases in Banach Spaces \rom1, Springer–Verlag, Berlin, Heidelberg, New York (1970) \ref\key 14 ––––, Bases in Banach Spaces \rom2, Springer–Verlag, Berlin, Heidelberg, New York (1981) \ref\key 15
  • A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw–Hill, New York (1978) \ref\key 16
  • Y. Yilmaz, Structural properties of some function spaces , Nonlinear Anal., 59(2004), 959–971 \ref\key 17 ––––, Characterizations of some operator spaces by relative adjoint operators , Nonlinear Anal., 65 (2006), 1833–1842