Strict topologies and Banach-Steinhaus type theorems

Marian Nowak

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 4, page 563-568
  • ISSN: 0010-2628

Abstract

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Let X be a completely regular Hausdorff space, E a real Banach space, and let C b ( X , E ) be the space of all E -valued bounded continuous functions on X . We study linear operators from C b ( X , E ) endowed with the strict topologies β z ( z = σ , τ , ∞ , g ) to a real Banach space ( Y , ∥ · ∥ Y ) . In particular, we derive Banach-Steinhaus type theorems for ( β z , ∥ · ∥ Y ) continuous linear operators from C b ( X , E ) to Y . Moreover, we study σ -additive and τ -additive operators from C b ( X , E ) to Y .

How to cite

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Nowak, Marian. "Strict topologies and Banach-Steinhaus type theorems." Commentationes Mathematicae Universitatis Carolinae 50.4 (2009): 563-568. <http://eudml.org/doc/35130>.

@article{Nowak2009,
abstract = {Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies $\beta _z$$(z=\sigma ,\tau ,\infty ,g)$ to a real Banach space $(Y,\Vert \cdot \Vert _Y)$. In particular, we derive Banach-Steinhaus type theorems for $(\beta _z,\Vert \cdot \Vert _Y)$ continuous linear operators from $C_b(X,E)$ to $Y$. Moreover, we study $\sigma $-additive and $\tau $-additive operators from $C_b(X,E)$ to $Y$.},
author = {Nowak, Marian},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector-valued continuous functions; strict topologies; locally solid topologies; Dini-topologies; strong Mackey space; $\sigma $-additive operators; $\tau $-additive operators; vector-valued continuous function; strict topology; locally solid topology; Dini topology; strong Mackey space; -additive operator; -additive operator},
language = {eng},
number = {4},
pages = {563-568},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Strict topologies and Banach-Steinhaus type theorems},
url = {http://eudml.org/doc/35130},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Nowak, Marian
TI - Strict topologies and Banach-Steinhaus type theorems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 4
SP - 563
EP - 568
AB - Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies $\beta _z$$(z=\sigma ,\tau ,\infty ,g)$ to a real Banach space $(Y,\Vert \cdot \Vert _Y)$. In particular, we derive Banach-Steinhaus type theorems for $(\beta _z,\Vert \cdot \Vert _Y)$ continuous linear operators from $C_b(X,E)$ to $Y$. Moreover, we study $\sigma $-additive and $\tau $-additive operators from $C_b(X,E)$ to $Y$.
LA - eng
KW - vector-valued continuous functions; strict topologies; locally solid topologies; Dini-topologies; strong Mackey space; $\sigma $-additive operators; $\tau $-additive operators; vector-valued continuous function; strict topology; locally solid topology; Dini topology; strong Mackey space; -additive operator; -additive operator
UR - http://eudml.org/doc/35130
ER -

References

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  7. Khurana S.S., Vielma J., 10.1006/jmaa.1995.1353, J. Math. Anal. Appl. 195 (1995), 251--260. Zbl0854.46032MR1352821DOI10.1006/jmaa.1995.1353
  8. Nowak M., Rzepka A., Locally solid topologies on spaces of vector-valued continuous functions, Comment. Math. Univ. Carolinae 43 (2002), no. 3, 473--483. Zbl1068.46023MR1920522
  9. Schaefer H., Zhang X.-D., On the Vitali-Hahn-Saks theorem, Oper. Theory Adv. Appl., 75, Birkhäuser, Basel, 1995, pp. 289--297. Zbl0830.28007MR1322508

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