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The extension of the Krein-Šmulian theorem for order-continuous Banach lattices

Antonio S. Granero, Marcos Sánchez (2008)

Banach Center Publications

If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) d ̂ ( c o ¯ w * ( K ) , X ) ≤ 2 d ̂ ( K , X ) and, if K ∩ X is w*-dense in K, then d ̂ ( c o ¯ w * ( K ) , X ) = d ̂ ( K , X ) ; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then d ̂ ( c o ¯ w * ( K ) , X ) = d ̂ ( K , X ) ; (iii) if X has a 1-symmetric basis, then d ̂ ( c o ¯ w * ( K ) , X ) = d ̂ ( K , X ) .

The fixed point property in a Banach space isomorphic to c 0

Costas Poulios (2014)

Commentationes Mathematicae Universitatis Carolinae

We consider a Banach space, which comes naturally from c 0 and it appears in the literature, and we prove that this space has the fixed point property for non-expansive mappings defined on weakly compact, convex sets.

The fixed point property in Musielak-Orlicz sequence spaces

Harold Bevan Thompson, Yunan Cui (2001)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we give necessary and sufficient conditions for a point in a Musielak-Orlicz sequence space equipped with the Orlicz norm to be an H-point. We give necessary and sufficient conditions for a Musielak-Orlicz sequence space equipped with the Orlicz norm to have the Kadec-Klee property, the uniform Kadec-Klee property and to be nearly uniformly convex. We show that a Musielak-Orlicz sequence space equipped with the Orlicz norm has the fixed point property if and only if it is reflexive....

The Fourier transform in Lebesgue spaces

Erik Talvila (2025)

Czechoslovak Mathematical Journal

For each f ∈ L p ( ℝ ) ( 1 ≤ p < ∞ ) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each p , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to L p ( ℝ ) . There is an exchange theorem and inversion in norm.

The functor σ²X

Stevo Todorčević (1995)

Studia Mathematica

We disprove the existence of a universal object in several classes of spaces including the class of weakly Lindelöf Banach spaces.

The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces

Harold Bennett, David Lutzer (2013)

Fundamenta Mathematicae

We examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of σ-isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. We show that any monotonically normal space with property * or with a σ-isolated network must be hereditarily paracompact, so that property * and the Gruenhage property are equivalent in monotonically normal spaces....

The Hyers-Ulam-Aoki Type Stability of Some Functional Equations on Banach Lattices

Nutefe Kwami Agbeko (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

In Agbeko (2012) the Hyers-Ulam-Aoki stability problem was posed in Banach lattice environments with the addition in the Cauchy functional equation replaced by supremum. In the present note we restate the problem so that it relates not only to supremum but also to infimum and their various combinations. We then propose some sufficient conditions which guarantee its solution.

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