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Analysis in Banach spaces
Pernecká, Eva ; Hájek, Petr (advisor) ; Johanis, Michal (referee) ; Godefroy, Gilles (referee)
The thesis consists of two papers and one preprint. The two papers are de- voted to the approximation properties of Lipschitz-free spaces. In the first pa- per we prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. In particular, the Lipschitz-free space over a closed subset of Rn has the bounded approximation property. We also show that the Lipschitz-free spaces over ℓ1 and over ℓn 1 admit a monotone finite-dimensional Schauder decomposition. In the second paper we improve this work and obtain even a Schauder basis in the Lipschitz-free spaces over ℓ1 and ℓn 1 . The topic of the preprint is rigidity of ℓ∞ and ℓn ∞ with respect to uniformly differentiable map- pings. Our main result is a non-linear analogy of the classical result on rigidity of ℓ∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in ℓ∞ due to Phillips. 1
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Lipschitz-free spaces
Langr, Ondřej ; Cúth, Marek (advisor) ; Johanis, Michal (referee)
In this work we deal with basic properties of Lipschitz-free space. In the first part we especially show how these spaces are constructed and we show that they are characterized by "Universal property". In the second part we give an explicit formula for the calculation of the norm of an element in the general Lipschitz- free space over metric space containing four points. It looks that this formula is nowhere published, therefore this is probably the original result of this work. 1
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Analysis in Banach spaces
Pernecká, Eva ; Hájek, Petr (advisor) ; Johanis, Michal (referee) ; Godefroy, Gilles (referee)
The thesis consists of two papers and one preprint. The two papers are de- voted to the approximation properties of Lipschitz-free spaces. In the first pa- per we prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. In particular, the Lipschitz-free space over a closed subset of Rn has the bounded approximation property. We also show that the Lipschitz-free spaces over ℓ1 and over ℓn 1 admit a monotone finite-dimensional Schauder decomposition. In the second paper we improve this work and obtain even a Schauder basis in the Lipschitz-free spaces over ℓ1 and ℓn 1 . The topic of the preprint is rigidity of ℓ∞ and ℓn ∞ with respect to uniformly differentiable map- pings. Our main result is a non-linear analogy of the classical result on rigidity of ℓ∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in ℓ∞ due to Phillips. 1
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