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4 Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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Orthogonally complemented hyperplanes in Orlicz function spaces

Chen S., Hudzik H., Wang Y.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr spec.

71-82

In this paper, necessary and sufficient conditions for a closed hyperplane in Orlicz space to be generalized orthogonally complemented are given for both the Orlicz and the Luxemburg norm. The concept of strongly generalized orthogonally complemented subspace in Banach space is defined and criteria for such subspaces in Orlicz space for both norms are given.

On the structure of the set of best ||.||Φ-approximants

Mazzone F.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2002

[Z] 42(1)

87-102

Certain properties of Fi-approximants and || || (fi)-approximants were studied by Landers and Rogge, where || || (fi) is the Luxemburg norm. In particular, they investigated the existence of best ||.|| (fi)-approximants and the structure of the . set of best ||.||(fi)-approximants. These authors proved that the set of best ||.||(fi)-approximants of f given a Fi-closed lattice C is a lattice. In this paper we show that this result does not hold if we consider the Orlicz norm in place of the Luxemburg norm. Furthermore, we see that for a large class of functions Fi and measurable spaces the following statements are equivalent: 1) the set of all best || || (fi)-approximants to f in C is a lattice, for every Fi-closed lattice C and f L_fi. 2) (L_fi,,||.||fi) = (L_p,m||.||_p), for some m > 0 and 1 < p < oo.

On some local and global geometry of Orlicz sequence spaces endowed with the Orlicz norm

Hao C., Li Y., Wang T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2000

[Z] 40

69-91

A critetion of URWC point in Orlicz sequence spaces endowed with the Orlicz norm is given. As a corollary, we get a criterion in order that a space has LURWC property.

Continuity of the metric projection in Orlicz space

Teng Y., Wang T., Li M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1999

[Z] 39

109-116

Let (f(n)) be a sequence of functions converging in norm to f in some rotund Orlicz function or se-quence space endowed with the Luxemburg norm or the Orlicz norm, and let (C(n)) be a sequence of convex sets satisfying some condition and tending in suitable way to a ser C. Then the best norm approximation of f(n) with respect to C(n) converges in norm to the best approximation of f with respect to C.