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Special symmetries of Banach spaces isomorphic to Hilbert spaces

100%
Talponen J.
Studia Mathematica
|
2010
|
tom 200
|
nr 1
31-40
EN
We characterize Hilbert spaces among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical: If X is a real Banach space isomorphic to a Hilbert space and convex-transitive with respect to the isometric finite-dimensional perturbations of the identity, then X is already isometric to a Hilbert space.
2
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A Natural Class of Sequential Banach Spaces

100%
Talponen J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 2
185-196
EN
We introduce and study a natural class of variable exponent $ℓ^{p}$ spaces, which generalizes the classical spaces $ℓ^{p}$ and c₀. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. Some geometric examples are constructed by using these spaces.
3
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Directionally Euclidean structures of Banach spaces

100%
Talponen J.
Studia Mathematica
|
2011
|
tom 202
|
nr 2
191-203
EN
We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term 'asymptotical' refers to the fact that we take 'lim sup' over finite-dimensional subspaces. This leads to isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem is given. We also discuss the stability of the family of ellipsoids as the dimension and geometry vary. The methods exploit ultrafilter techniques and we also apply them in conjunction with finite Auerbach bases to study the convexity properties of the duality mappings.
4
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Diameter 2 properties and convexity

45%
Abrahamsen T., Hájek P., Nygaard O., Talponen J., Troyanski S.
Studia Mathematica
|
2016
|
tom 232
|
nr 3
227-242
EN
We present an equivalent midpoint locally uniformly rotund (MLUR) renorming of C[0,1] with the diameter 2 property (D2P), i.e. every non-empty relatively weakly open subset of the unit ball has diameter 2. An example of an MLUR space with the D2P and with convex combinations of slices of arbitrarily small diameter is also given.
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