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On the complexity of Hamel bases of infinite-dimensional Banach spaces

Lorenz Halbeisen — 2001

Colloquium Mathematicae

We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.

Families of almost disjoint Hamel bases.

Lorenz Halbeisen — 2005

Extracta Mathematicae

For infinite dimensional Banach spaces X we investigate the maximal size of a family of pairwise almost disjoint normalized Hamel bases of X, where two sets A and B are said to be almost disjoint if the cardinality of A ∩ B is smaller than the cardinality of either A or B.

Ramseyan ultrafilters

Lorenz Halbeisen — 2001

Fundamenta Mathematicae

We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we...

The Josephus problem

Lorenz HalbeisenNorbert Hungerbühler — 1997

Journal de théorie des nombres de Bordeaux

We give explicit non-recursive formulas to compute the Josephus-numbers j ( n , 2 , i ) and j ( n , 3 , i ) and explicit upper and lower bounds for j ( n , k , i ) (where k 4 ) which differ by 2 k - 2 (for k = 4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j ( n , k , i ) which is based upon the mentioned bounds.

On bases in Banach spaces

We investigate various kinds of bases in infinite-dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in as well as in separable Banach spaces.

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