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Copies of the sequence space ω in F-lattices with applications to Musielak−Orlicz spaces

Wójtowicz M., Wiśniewska H.

Commentationes Mathematicae

2016

Vol 56, No. 1

103--117

Let E be a fixed real function F-space, i.e., E is an order ideal in L0(S,Σ,μ) endowed with a monotone F-norm ∥∥ under which E is topologically complete. We prove that E contains an isomorphic (topological) copy of ω, the space of all sequences, if and only if E contains a lattice-topological copy W of ω. If E is additionally discrete, we obtain a much stronger result: W can be a projection band; in particular, E contains a~complemented copy of ω. This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of ω by a Musielak−Orlicz space is characterized as follows. (1) A sequence space ℓΦ, where Φ=(φn), contains a copy of ω iff infn∈Nφn(∞)=0, where φn(∞)=limt→∞φn(t). (2) If the measure μ is atomless, then ω embeds isomorphically into LM(μ) iff the function M∞ is positive and bounded on some set A∈Σ of positive and finite measure, where M∞ (s)=limn→∞M(n,s), s∈S. In particular, (1)' ℓφ does not contain any copy of ω, and (2)' Lφ(μ), with μ atomless, contains a~copy W of ω iff φ is bounded, and every such copy W is uncomplemented in Lφ(μ).

On maps which preserve equality of distance in F*-spaces

Tan D.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 3-4

225-230

We prove that every map T between two F*-spaces which preserves equality of distance and satisfies T(0) = 0 is linear.

Refining two Marciszewski's constructions related to the Sierpiński-Kuratowski technique

Nunez Garcia J.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 2

157-161

Refining two Marciszewski's constructions, we present an example of a line-free group with no continuous functions onto its own square, and we give a new proof showing that van Mill's space L [6] is not homeomorphic to L x Z for any nontrival Z.

Noncontinuous linear operators and independent linear topologies

Lipecki Z.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1998

[Z] 38

53-62

We establish the existence of badly noncontinuous injective linear operators, isomorphisms and involutions between topological linear spaces. As an application, a metrizable linear topology independent of a given one is exhibited.