On some properties of Musielak-Orlicz sequence spaces

Isaac V. Shragin. "On some properties of Musielak-Orlicz sequence spaces." Commentationes Mathematicae 48.2 (2008): null. <http://eudml.org/doc/292012>.

@article{IsaacV2008,
abstract = {We consider a nontrivial vector space $X$ and a semimodular $M\colon X[0, \infty ]$ with property: $(\forall \ x \in X) (\exists \ \alpha 0)\ M () \infty $ (in other words, $M$ is normal (i.e. $(\forall \ x\in X \setminus \lbrace 0\rbrace ) (\exists \alpha 0)\ M () 0)$ pregenfunction). The function $M$ generates in $X$ a metric $d$ with \[ d(x, y) := inf \lbrace a 0: M (a^\{-1\} (x-y)) \le a\rbrace . \] At the same time $M$ generates a metric $\rho $ in Musielak-Orlicz sequence space $l_M$, namely \[ \rho (\varphi , \psi ) := inf \lbrace a 0 : I(a^\{-1\} (\varphi - \psi )) \le a\rbrace \] with $I(\varphi ) = \sum _\{n \ge 1\} M (\varphi φ(n))$. It is proved that the space $(l_M,\rho )$ is complete if and only if the space $(X, d)$ is complete. We consider also the closed subspace $G_M \subset l_M$ of sequences $\varphi = \lbrace \varphi (n)\rbrace $ such that $(\forall \alpha 0) (\exists m \in N) \sum _\{n\ge m\} M(\alpha \varphi (n)) \infty $ and prove that $(G_M ,\rho )$ is separable if and only if $(X, d)$ is the same. Several examples are considered.},
author = {Isaac V. Shragin},
journal = {Commentationes Mathematicae},
keywords = {normal pregenfunction; Musielak-Orlicz sequence space; completeness; separability},
language = {eng},
number = {2},
pages = {null},
title = {On some properties of Musielak-Orlicz sequence spaces},
url = {http://eudml.org/doc/292012},
volume = {48},
year = {2008},
}

TY - JOUR
AU - Isaac V. Shragin
TI - On some properties of Musielak-Orlicz sequence spaces
JO - Commentationes Mathematicae
PY - 2008
VL - 48
IS - 2
SP - null
AB - We consider a nontrivial vector space $X$ and a semimodular $M\colon X[0, \infty ]$ with property: $(\forall \ x \in X) (\exists \ \alpha 0)\ M () \infty $ (in other words, $M$ is normal (i.e. $(\forall \ x\in X \setminus \lbrace 0\rbrace ) (\exists \alpha 0)\ M () 0)$ pregenfunction). The function $M$ generates in $X$ a metric $d$ with \[ d(x, y) := inf \lbrace a 0: M (a^{-1} (x-y)) \le a\rbrace . \] At the same time $M$ generates a metric $\rho $ in Musielak-Orlicz sequence space $l_M$, namely \[ \rho (\varphi , \psi ) := inf \lbrace a 0 : I(a^{-1} (\varphi - \psi )) \le a\rbrace \] with $I(\varphi ) = \sum _{n \ge 1} M (\varphi φ(n))$. It is proved that the space $(l_M,\rho )$ is complete if and only if the space $(X, d)$ is complete. We consider also the closed subspace $G_M \subset l_M$ of sequences $\varphi = \lbrace \varphi (n)\rbrace $ such that $(\forall \alpha 0) (\exists m \in N) \sum _{n\ge m} M(\alpha \varphi (n)) \infty $ and prove that $(G_M ,\rho )$ is separable if and only if $(X, d)$ is the same. Several examples are considered.
LA - eng
KW - normal pregenfunction; Musielak-Orlicz sequence space; completeness; separability
UR - http://eudml.org/doc/292012
ER -