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On some properties of Musielak-Orlicz sequence spaces

Isaac V. Shragin — 2008

Commentationes Mathematicae

We consider a nontrivial vector space X and a semimodular M : X [ 0 , ] with property: ( x X ) ( α 0 ) M ( ) (in other words, M is normal (i.e. ( x X { 0 } ) ( α 0 ) M ( ) 0 ) pregenfunction). The function M generates in X a metric d with d ( x , y ) : = i n f { a 0 : M ( a - 1 ( x - y ) ) a } . At the same time M generates a metric ρ in Musielak-Orlicz sequence space l M , namely ρ ( ϕ , ψ ) : = i n f { a 0 : I ( a - 1 ( ϕ - ψ ) ) a } with I ( ϕ ) = n 1 M ( ϕ φ ( n ) ) . It is proved that the space ( l M , ρ ) is complete if and only if the space ( X , d ) is complete. We consider also the closed subspace G M l M of sequences ϕ = { ϕ ( n ) } such that ( α 0 ) ( m N ) n m M ( α ϕ ( n ) ) and prove that ( G M , ρ ) is separable if and only if ( X , d ) is the same. Several examples are...

On the boundedness of sets in Musielak-Orlicz spaces

Isaac V. Shragin — 2013

Commentationes Mathematicae

The notions of metrical and topological vector boundedness of sets are considered in Musielak-Orlicz spaces. The space is called right if both these notions are equivalent. Necessary conditions of the rightness are established, and certain sufficient conditions are found.

Corrigendum to the paper...

Isaac V. Shragin — 2009

Commentationes Mathematicae

Corrections of misprints in the article of I.V. Shragin On a measurability of a superposition, which defines nonlinear integral Musielak operator (Comment. Mathem. Tomus Specialis in honorem Juliani Musialak, Warszawa 2004).

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