On the quotient shapes of topological spaces

Uglešić, Nikica

doi:10.1016/j.topol.2018.02.022

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…In [13], several well known concrete categories were considered and many examples are given which show that the quotient shape theory yields classifications strictly coarser than those by isomorphisms. In [14] and [15] were considered the quotient shapes of (purely algebraic, topological and normed -the category N F ) vectorial spaces and topological spaces, respectively. In paper [16], we have continued the studying of quotient shapes of normed vectorial spaces of [14], Section 4.1, primarily and separately focused to the well known l p and L p spaces.…”

Section: Introductionmentioning

confidence: 99%

On the duals of normed spaces and quotient shapes

Uglesic

2019

Preprint

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Some properties of the (normed) dual Hom-functor D and its iterations D n are exhibited. For instance: D turns every canonical embedding (in the second dual space) into a retraction (of the third dual onto the first one); D rises the countably infinite (algebraic) dimension only; D does not change the finite quotient shape type. By means of that, the finite quotient shape classification of normed vectorial spaces is completely solved. As a consequence, two extension type theorems are derived.

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Section: Introductionmentioning

confidence: 99%

On the duals of normed spaces and quotient shapes

Uglesic

2019

Preprint

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“…In [12], several well known concrete categories were considered and many examples are given which show that the quotient shape theory yields classifications strictly coarser than those by isomorphisms. In [13] and [14] were considered the quotient shapes of (purely algebraic, topological and normed) vectorial spaces and topological spaces, respectively. In the recent paper [15], we have continued the studying of quotient shapes of normed vectorial spaces of [13], Section 4.1, primarily and separately focused to the well known l p and L p spaces and to the Sobolev spaces W (k) p (Ω n ) (of all real functions on Ω n having their supports in a domain Ω n and all partial derivatives up to order k continuous).…”

Section: Introductionmentioning

confidence: 99%

The quotient shapes of normed spaces and application

Uglešić¹

2019

Preprint

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The quotient shape types of normed vectorial spaces (over the same field) with respect to Banach spaces reduce to those of Banach spaces. The finite quotient shape type of normed spaces is an invariant of the (algebraic) dimension, but not conversely. The converse holds for separable normed spaces as well as for the bidual-like spaces (isomorphic to their second dual spaces). As a consequence, the Hilbert space l 2 , or even its (countably dimensional, unitary) direct sum subspace may represent the unique quotient shape type of all 2 ℵ0 -dimensional normed spaces. An application yields two extension type theorems.

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“…In [13], several well known concrete categories were considered and many examples are given which show that the quotient shape theory yields classifications strictly coarser than those by isomorphisms. In [14] and [15] were considered the quotient shapes of (purely algebraic, topological and normed) vectorial spaces and topological spaces, respectively. In this paper we continue the studying of quotient shapes of normed vectorial spaces of [Section 4.1,14], primarily focused to the well known l p and L p spaces.…”

Section: Introductionmentioning

confidence: 99%

The quotient shapes of lp and Lp spaces

Uglešić¹

2018

Rad Hrvatske akademije znanosti i umjetnosti

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All lp spaces (over the same field), p ̸ = ∞, have the finite quotient shape type of the Hilbert space l 2 . It is also the finite quotient shape type of all the subspaces lp(p ′ ), p < p ′ ≤ ∞, as well as of all their direct sum subspaces F N 0 (p ′ ), 1 ≤ p ′ ≤ ∞. Furthermore, their countable and finite quotient shape types coincide. Similarly, for a given positive integer, all Lp spaces (over the same field) have the finite quotient shape type of the Hilbert space L 2 , and their countable and finite quotient shape types coincide. Quite analogous facts hold true for the (special type of) Sobolev spaces (of all appropriate real functions).

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On the quotient shapes of topological spaces

Cited by 4 publications

References 3 publications

On the duals of normed spaces and quotient shapes

On the duals of normed spaces and quotient shapes

The quotient shapes of normed spaces and application

The quotient shapes of lp and Lp spaces

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