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Continuity of the identity embedding of Musielak-Orlicz sequence spaces. (English) Zbl 0645.46009
Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 427-437 (1987).
[For the entire collection see Zbl 0627.00012.]
Let X be a real linear space and let \({\mathcal X}\) be the space of all sequences of elements of X. A sequence \(\phi =(\phi_ n)\), \(\phi_ n:X\to [0,\infty]\) is said to be a \(\phi\)-function if the following conditions are satisfied for all \(n\in {\mathbb{N}}:\)
a) \(\phi_ n(0)=0\), \(\phi_ n(-x)=\phi_ n(x)\) for \(x\in X,\)
b) \(\lim_{u\to 0}\phi_ n(ux)=0\) for \(x\in X\) wth \(\phi_ n(x)<\infty,\)
c) \(\phi_ n(ux+vy)\leq \phi_ n(x)+\phi_ n(y)\) for x,y\(\in X\) and u,v\(\geq 0\) with \(u+v=1.\)
For a \(\phi\)-function \(\phi\) the functional \(I_{\phi}:{\mathcal X}\to [0,\infty]\) is defined by \(I_{\phi}(f)=\sum^{\infty}_{n=1}\phi_ n(f_ n)\) for \(f=(f_ n)\in {\mathcal X}\). A sequence (f(m)) of element of \({\mathcal X}\) is said to be \(I_{\phi}\)-convergent [resp. \(N_{\phi}\)- convergent] to \(f\in {\mathcal X}\) if \(\lim_{m\to \infty}I_{\phi}(a(f(m)- f))=0\) for some \(a>0\) [resp. for all \(a>0]\). The Musielak-Orlicz sequence space \(\ell^{\phi}\) is defined as the space of all \(f\in {\mathcal X}\) such that \(I_{\phi}(af)<\infty\) for some \(a>0\). It is well known [I. V. Shragin, Mat. Zametki 20, 681-692 (1976; Zbl 0349.46011)] that the inclusion \(\ell^{\phi}\subset \ell^{{\bar \psi}}\) holds if and only if there are numbers \(a,c,K>0\), \(n_ 0\in {\mathbb{N}}\) such that \(\sum^{\infty}_{n=n_ 0}\alpha_ n(a,c,K)<\infty\), where \(\alpha_ n(a,c,K)=\sup \{\psi_ n(cx):x\in P_ n(a,c,K)\}\) and \(P_ n(a,c,K)=\{x\in X:\phi_ n(x)\leq a\), \(\psi_ n(cx)>K\phi_ n(x)\}\). The author obtains conditions under which the identity embedding \(i:\ell^{\phi}\to \ell^{\psi}\) is continuous. A typical example of the theorems of the paper is as follows: \(\ell^{\phi}\subset \ell^{\psi}\) and \(I_{\phi}\)-convergence implies \(N_{\psi}\)- convergence if and only if \(\forall \epsilon >0\), \(\forall c>0\), \(\exists a>0\), \(\exists K>0\), \(\sum^{\infty}_{n=1}\alpha_ n(a,c,K)<\epsilon\).
Reviewer: T.Leiger

46A45 Sequence spaces (including Köthe sequence spaces)
46B25 Classical Banach spaces in the general theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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