Schauder bases and diametrically complete sets with empty interior

Budzyńska, Monika; Grzesik, Aleksandra; Kaczor, Wiesława; Kuczumow, Tadeusz

doi:10.1016/j.jmaa.2018.02.024

Cited by 6 publications

(7 citation statements)

References 5 publications

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“…In [6], the following results were proved. Now we are ready to prove two main theorems of this section.…”

Section: The Day Norm and The Second Kind Of Renormingsmentioning

confidence: 87%

“…We also have the following theorem ( [5]) which proof is partially based on ideas given in [11]. Theorem 3.6.…”

Section: Unconditionality and 1-unconditionality Of Bases The Non-stmentioning

confidence: 99%

“…The first part of this theorem was proved in [5], but for the convenience of the reader we recall it. So we start with the observation that by Theorem 2.4 we have X ⊂ l ∞ = (l 1 ) * = Y * with Y = l 1 .…”

Section: Unconditionality and 1-unconditionality Of Bases The Non-stmentioning

confidence: 99%

“…It is worth pointing out that the renormings used in this section were introduced in [5]. Moreover, this kind of renormings comes from the Karlovitz paper [16].…”

Section: The First Kind Of Renormingsmentioning

confidence: 99%

“…(see [6], [8] and [25]). Now let {e i } i be a normalized Schauder basis in (X, · ) and {e * i } i a sequence of biorthogonal functionals associated with this basis.…”

Section: The Day Norm and The Second Kind Of Renormingsmentioning

confidence: 99%

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A remark on the fixed point property of nonexpansive mappings

2018

J. Nonlinear Var. Anal.

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“…In [6], the following results were proved. Now we are ready to prove two main theorems of this section.…”

Section: The Day Norm and The Second Kind Of Renormingsmentioning

confidence: 87%

“…We also have the following theorem ( [5]) which proof is partially based on ideas given in [11]. Theorem 3.6.…”

Section: Unconditionality and 1-unconditionality Of Bases The Non-stmentioning

confidence: 99%

Section: Unconditionality and 1-unconditionality Of Bases The Non-stmentioning

confidence: 99%

“…It is worth pointing out that the renormings used in this section were introduced in [5]. Moreover, this kind of renormings comes from the Karlovitz paper [16].…”

Section: The First Kind Of Renormingsmentioning

confidence: 99%

“…(see [6], [8] and [25]). Now let {e i } i be a normalized Schauder basis in (X, · ) and {e * i } i a sequence of biorthogonal functionals associated with this basis.…”

Section: The Day Norm and The Second Kind Of Renormingsmentioning

confidence: 99%

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A remark on the fixed point property of nonexpansive mappings

2018

J. Nonlinear Var. Anal.

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Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

Kaczor¹,

Kuczumow²,

Reich

et al. 2023

Results Math

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We show that each infinite-dimensional reflexive Banach space $$(X,\left\| \cdot \right\| _X)$$ ( X , · X ) has an equivalent norm $$\left\| \cdot \right\| _{X,0}$$ · X , 0 such that $$(X,\left\| \cdot \right\| _{X,0})$$ ( X , · X , 0 ) is LUR and contains a diametrically complete set with empty interior. We also prove that after a suitable equivalent renorming, the Banach space $$C([0,1],{\mathbb {R}})$$ C ( [ 0 , 1 ] , R ) contains a constant width set with empty interior.

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Complete sets in normed linear spaces

Martini

2023

Banach J. Math. Anal.

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A bounded subset of a (finite or infinite dimensional) normed linear space is said to be complete (or diametrically complete) if it cannot be enlarged without increasing its diameter. Any bounded subset A of a normed linear space is contained in a complete set having the same diameter, which is called a completion of A. We survey characterizations, basic properties, facts about structure of the interior and boundary, and the asymmetry of complete sets. Different methods to obtain completions of bounded sets are presented. Moreover, the structure of the space of complete sets endowed with the Hausdorff metric and relations of this set family to related set families and notions are discussed. For example, we mean here sets of constant width, balls, reduced sets, sets of constant diameter, and sets of constant radius.

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Schauder bases and diametrically complete sets with empty interior

Cited by 6 publications

References 5 publications

A remark on the fixed point property of nonexpansive mappings

A remark on the fixed point property of nonexpansive mappings

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

Complete sets in normed linear spaces

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