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On convex functions in c0(w1).

Petr Hájek (1996)

Collectanea Mathematica

It is proved that no convex and Fréchet differentiable function on c0(w1), whose derivative is locally uniformly continuous, attains its minimum at a unique point.

On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček (1997)

Commentationes Mathematicae Universitatis Carolinae

We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux differentiable bump function. In particular, our result implies the following theorem: If d is a distance function determined by a closed subset A of a Banach space X with a uniformly Gâteaux differentiable norm, then the set of points of X A at which d is not Gâteaux differentiable is not only a first category set, but it is even σ -porous...

On Fréchet differentiability of convex functions on Banach spaces

Wee-Kee Tang (1995)

Commentationes Mathematicae Universitatis Carolinae

Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function f defined on a separable Banach space are studied. The conditions are in terms of a majorization of f by a C 1 -smooth function, separability of the boundary for f or an approximation of f by Fréchet smooth convex functions.

On Gateaux differentiable bump functions

Francisco Hernández, Stanimir Troyanski (1996)

Studia Mathematica

It is shown that the order of Gateaux smoothness of bump functions on a wide class of Banach spaces with unconditional basis is not better than that of Fréchet differentiability. It is proved as well that in the separable case this order for Banach lattices satisfying a lower p-estimate for 1≤ p < 2 can be only slightly better.

On integrability in F-spaces

Mikhail Popov (1994)

Studia Mathematica

Some usual and unusual properties of the Riemann integral for functions x : [a,b] → X where X is an F-space are investigated. In particular, a continuous integrable l p -valued function (0 < p < 1) with non-differentiable integral function is constructed. For some class of quasi-Banach spaces X it is proved that the set of all X-valued functions with zero derivative is dense in the space of all continuous functions, and for any two continuous functions x and y there is a sequence of differentiable...

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