|
|
Basic properties of p-Banach spaces
Kubíček, David ; Cúth, Marek (advisor) ; Johanis, Michal (referee)
In this thesis we recall the notion of a quasi-norm and a p-norm. We mention the Aoki-Rolewicz theorem which connects these two notions. We deal with generalizations of selected results from Banach spaces to p-Banach spaces for 0 < p ≤ 1. We study the class of Lp(µ) spaces. Namely, we explore their basic properties, their dual spaces and nonlinear structure. 1
|
|
| |
|
|
Finitely additive measures and their docompositions
Zindulka, Mikuláš ; Cúth, Marek (advisor) ; Johanis, Michal (referee)
We define the notion of a finitely additive measure on a σ-algebra. We prove that a bounded finitely additive measure can be uniquely represented as a sum of a "σ-additive part" and a "purely finitely additive part" and that it also has a decomposition similar to the Lebesgue decomposition for σ-additive measures. Bounded finitely additive measures defined on the Borel σ-algebra form a normed linear space and those that are zero on Lebesgue null sets form its subspace. We show that the former one is isometrically isomorphic to the dual space of the space of bounded Borel functions and the latter one is isometrically isomorphic to the dual space of the space of essentially bounded functions. 1
|
|
|
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
|
|
|
Semiconvex functions and its differences
Kryštof, Václav ; Zajíček, Luděk (advisor) ; Johanis, Michal (referee)
The main result of this thesis is that we prove certain versions of Ilmanen's lemmma. That means - given semiconvex (or locally semiconvex) function f1 and semiconcave (or locally semiconcave) function f2 such that f1 ≤ f2 we find a function f such that f1 ≤ f ≤ f2 and f is both semiconvex and semiconcave (or locally uniformly differentiable). We also give characterization (via a new variation) of those functions which are the difference of two ω-nondecreasing functions 1
|
|
|
Basic sequences in Banach spaces
Zindulka, Mikuláš ; Kalenda, Ondřej (advisor) ; Johanis, Michal (referee)
An ordering on bases in Banach spaces is defined as a natural generalization of the notion of equivalence. Its theory is developed with emphasis on its behavior with respect to shrinking and boundedly-complete bases. We prove that a bounded operator mapping a shrinking basis to a boundedly-complete one is weakly compact. A well-known result concerning the factorization of a weakly compact operator through a reflexive space is then reinterpreted in terms of the ordering. Next, we introduce a class of Banach spaces whose norm is constructed from a given two-dimensional norm N. We prove that any such space XN is isomorphic to an Orlicz sequence space. A key step in obtaining this correspondence is to describe the unit circle in the norm N with a convex function ϕ. The canonical unit vectors form a basis of a subspace YN of XN . We characterize the equivalence of these bases and the situation when the basis is boundedly-complete. The criteria are formulated in terms of the norm N and the function ϕ. 1
|
|
|
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
|
|
|
Finitely additive measures and their docompositions
Zindulka, Mikuláš ; Cúth, Marek (advisor) ; Johanis, Michal (referee)
We define the notion of a finitely additive measure on a σ-algebra. We prove that a bounded finitely additive measure can be uniquely represented as a sum of a "σ-additive part" and a "purely finitely additive part" and that it also has a decomposition similar to the Lebesgue decomposition for σ-additive measures. Bounded finitely additive measures defined on the Borel σ-algebra form a normed linear space and those that are zero on Lebesgue null sets form its subspace. We show that the former one is isometrically isomorphic to the dual space of the space of bounded Borel functions and the latter one is isometrically isomorphic to the dual space of the space of essentially bounded functions. 1
|
|
|
Semiconvex functions and its differences
Kryštof, Václav ; Zajíček, Luděk (advisor) ; Johanis, Michal (referee)
The main result of this thesis is that we prove certain versions of Ilmanen's lemmma. That means - given semiconvex (or locally semiconvex) function f1 and semiconcave (or locally semiconcave) function f2 such that f1 ≤ f2 we find a function f such that f1 ≤ f ≤ f2 and f is both semiconvex and semiconcave (or locally uniformly differentiable). We also give characterization (via a new variation) of those functions which are the difference of two ω-nondecreasing functions 1
|
|
|
Bisectors
Los, Tomáš ; Johanis, Michal (advisor) ; Kalenda, Ondřej (referee)
This work deals with the study of bisectors (i.e. sets of points of equal distance from two given points) and the impact of their shape on the shape of the unit ball. It is known that if each bisector of two antipodal points on the sphere of a normed linear space lies in a hyperplane, then the norm is an inner product norm (for a special case of norm in R2 it is proved in Theorem 18). Here we generalise this statement in R2 for the case of (a priori) non-symmetric unit ball. In particular, we show that if the set of points x in the unit sphere, such that the bisector of x and −x is a line, has non-empty interior with respect to the sphere and the sphere is smooth, then the unit sphere is an ellipse centred at the origin. The work is based on the preprint [1]. 1
|
guest ::
