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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

##### MSC:
 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.)
##### Keywords:
Musielak-Orlicz sequence space; embedding
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