Dichotomies for ${𝐂}_0\left(X\right)$ and ${𝐂}_b\left(X\right)$ spaces

Głąb, Szymon, and Strobin, Filip. "Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces." Czechoslovak Mathematical Journal 63.1 (2013): 91-105. <http://eudml.org/doc/252460>.

@article{Głąb2013,
abstract = {Jachymski showed that the set \[ \bigg \lbrace (x,y)\in \{\bf c\}\_0\times \{\bf c\}\_0\colon \bigg (\sum \_\{i=1\}^n \alpha (i)x(i)y(i)\bigg )\_\{n=1\}^\infty \text\{is bounded\}\bigg \rbrace \] is either a meager subset of $\{\bf c\}_0\times \{\bf c\}_0$ or is equal to $\{\bf c\}_0\times \{\bf c\}_0$. In the paper we generalize this result by considering more general spaces than $\{\bf c\}_0$, namely $\{\bf C\}_0(X)$, the space of all continuous functions which vanish at infinity, and $\{\bf C\}_b(X)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma $-porosity.},
author = {Głąb, Szymon, Strobin, Filip},
journal = {Czechoslovak Mathematical Journal},
keywords = {continuous function; integration; Baire category; porosity; Banach space; continuous function; strongly ball porosity; -lower porosity},
language = {eng},
number = {1},
pages = {91-105},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Dichotomies for $\{\bf C\}_0(X)$ and $\{\bf C\}_b(X)$ spaces},
url = {http://eudml.org/doc/252460},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Głąb, Szymon
AU - Strobin, Filip
TI - Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 91
EP - 105
AB - Jachymski showed that the set \[ \bigg \lbrace (x,y)\in {\bf c}_0\times {\bf c}_0\colon \bigg (\sum _{i=1}^n \alpha (i)x(i)y(i)\bigg )_{n=1}^\infty \text{is bounded}\bigg \rbrace \] is either a meager subset of ${\bf c}_0\times {\bf c}_0$ or is equal to ${\bf c}_0\times {\bf c}_0$. In the paper we generalize this result by considering more general spaces than ${\bf c}_0$, namely ${\bf C}_0(X)$, the space of all continuous functions which vanish at infinity, and ${\bf C}_b(X)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma $-porosity.
LA - eng
KW - continuous function; integration; Baire category; porosity; Banach space; continuous function; strongly ball porosity; -lower porosity
UR - http://eudml.org/doc/252460
ER -