Mappings of Baire spaces into function spaces and Kadeč renorming

We investigate non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is σ -discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces C(K) with a σ -discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina concerning Borel structures and Kadec renormings in Banach spaces.

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“…[22], but it is not clear if the same is true for the (weak-norm)-perfectness. Some pertinent results were obtained by Oncina [24].…”

Section: The (Weak-norm)-λ-spacesmentioning

confidence: 78%

“…Section 2.5. To that end we shall need a variant of a "discretization" lemma used by Haydon [12] and Namioka and Pol [22], cf. also [3,Lemma 4.3].…”

Section: Aleksandrov-urysohn Compacta With (Weak-norm)-perfect Functimentioning

confidence: 99%

On Banach spaces whose norm-open sets are Fσ-sets in the weak topology

Marciszewski

Pol

2009

Journal of Mathematical Analysis and Applications

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“…Our counter-example is modelled on that of Namioka and Pol [15] which in turn, is based upon the following theorem of Kunen (see [16] for a proof). THEOREM 2.1.…”

Section: A Gc-space Without a Kadec Normmentioning

confidence: 99%

A dual differentiation space without an equivalent locally uniformly rotund norm

Kenderov

Moors

2004

J. Aust. Math. Soc.

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A Banach space (X, || • ||) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Frechet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.2000 Mathematics subject classification: primary 46B20; secondary 58C20.

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“…In the proof of Proposition 4.2, we shall use the following 'discretization' lemma, proved (in a slightly different form) in [7,8,17]. Proof of Proposition 4.2.…”

Section: A Function Space C(k) With Different Norm-borel and Weak-bormentioning

confidence: 99%