Wyniki wyszukiwania

1

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

100%

Mykhaylyuk V., Plichko A.

Colloquium Mathematicum

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2015

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tom 141

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nr 2

175-181

EN

We comment on a problem of Mazur from ``The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function $F_{y}(x): = f'_{y}(x,y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f''_{yx}$ exists. We construct a separately twice differentiable function whose partial derivative $f'_{x}$ is discontinuous with respect to the second variable on a set of positive measure. This solves the Mazur problem in the negative.

2

On local convexity of nonlinear mappings between Banach spaces

81%

Banakh I., Banakh T., Plichko A., Prykarpatsky A.

Open Mathematics

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2012

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tom 10

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nr 6

2264-2271

EN

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

3

On bases in Banach spaces

71%

Bartoszyński T., Džamonja M., Halbeisen L., Murtinová E., Plichko A.

Studia Mathematica

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2005

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tom 170

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nr 2

147-171

EN

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 $ℓ_{∞}$ as well as in separable Banach spaces.

4

On the Kunen-Shelah properties in Banach spaces

71%

Granero A., Jiménez M., Montesinos A., Moreno J., Plichko A.

Studia Mathematica

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2003

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tom 157

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nr 2

97-120

EN

We introduce and study the Kunen-Shelah properties $KS_{i}$, i = 0,1,...,7. Let us highlight some of our results for a Banach space X: (1) X* has a w*-nonseparable equivalent dual ball iff X has an ω-polyhedron (i.e., a bounded family $KS_{i}$, i = 0,1,...,7. Let us highlight some of our results for a Banach space X: (1) X* has a w*-nonseparable equivalent dual ball iff X has an ω-polyhedron (i.e., a bounded family ${x_{i}}_{i<ω}$ such that $x_{j} ∉ \overline{co}(x_{i}: i ∈ ω∖{j}})$ for every j ∈ ω) iff X has an uncountable bounded almost biorthogonal system (UBABS) of type η for some η ∈ [0,1) (i.e., a bounded family ${(x_{α},f_{α})}_{1≤α<ω} ⊂ X × X*$ such that $f_{α}(x_{α}) = 1$ and $|f_{α}(x_{β})| ≤ η$ if α ≠ β); (2) if X has an uncountable ω-independent system then X has an UBABS of type η for every η ∈ (0,1); (3) if X does not have the property (C) of Corson, then X has an ω-polyhedron; (4) X has no ω-polyhedron iff X has no convex right-separated ω-family (i.e., a bounded family ${x_{i}}_{i<ω}$ such that $x_{j} ∉ \overline{co}({x_{i}: j < i < ω})$ for every j ∈ ω) iff every w*-closed convex subset of X* is w*-separable iff every convex subset of X* is w*-separable iff μ(X) = 1, μ(X) being the Finet-Godefroy index of X (see [1]).

Ograniczanie wyników

2 Studia Mathematica

1 Colloquium Mathematicum

1 Open Mathematics

4 Plichko A.

1 Banakh I.

1 Banakh T.

1 Bartoszyński T.

1 Džamonja M.

1 Granero A.

1 Halbeisen L.

1 Jiménez M.

1 Montesinos A.

1 Moreno J.

1 Murtinová E.

1 Mykhaylyuk V.

1 Prykarpatsky A.

1 2015

1 2012

1 2005

1 2003

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

On local convexity of nonlinear mappings between Banach spaces

On bases in Banach spaces

On the Kunen-Shelah properties in Banach spaces