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30101 Pure Mathematics
2Locally convex space
2Quotient space
2Separable quotient problem
10103 Numerical and Computational Mathematics
10199 Other Mathematical Sciences
1Australian Digital Thesis
1Biorthogonal system
1Circle group
1Complemented subspace
1Dual space
1Frechet space
1Locally bounded space
1Markushevich base
1Nonconvex cone
1Normed linear spaces
1Nuclear space
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Vector optimization problems with nonconvex preferences

- Huang, N. J., Rubinov, Alex, Yang, Xiao

**Authors:**Huang, N. J. , Rubinov, Alex , Yang, Xiao**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 40, no. 4 (2008), p. 765-777**Full Text:**false**Reviewed:****Description:**In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given. © 2007 Springer Science+Business Media, Inc.**Description:**C1

Colocality and twisted sums of Banach spaces

- Jebreen, H. M., Jamjoom, F. B. H., Yost, David

**Authors:**Jebreen, H. M. , Jamjoom, F. B. H. , Yost, David**Date:**2006**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 323, no. 2 (2006), p. 864-875**Full Text:****Reviewed:****Description:**Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.**Description:**C1**Description:**2003001831

**Authors:**Jebreen, H. M. , Jamjoom, F. B. H. , Yost, David**Date:**2006**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 323, no. 2 (2006), p. 864-875**Full Text:****Reviewed:****Description:**Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.**Description:**C1**Description:**2003001831

Twisted sums with C(K) spaces

- Cabello Sanchez, Felix, Castillo, Jesus, Kalton, Nigel, Yost, David

**Authors:**Cabello Sanchez, Felix , Castillo, Jesus , Kalton, Nigel , Yost, David**Date:**2003**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 355, no. (2003), p. 4523-4541**Full Text:****Reviewed:****Description:**If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 -->**Description:**C1**Description:**2003002201

**Authors:**Cabello Sanchez, Felix , Castillo, Jesus , Kalton, Nigel , Yost, David**Date:**2003**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 355, no. (2003), p. 4523-4541**Full Text:****Reviewed:****Description:**If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 -->**Description:**C1**Description:**2003002201

Complemented and uncomplemented subspaces of Banach spaces

**Authors:**Vuong, Thi Minh Thu**Date:**2006**Type:**Text , Thesis , Masters**Full Text:****Description:**"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.**Description:**Master of Mathematical Sciences

**Authors:**Vuong, Thi Minh Thu**Date:**2006**Type:**Text , Thesis , Masters**Full Text:****Description:**"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.**Description:**Master of Mathematical Sciences

A topological group observation on the Banach-Mazur separable quotient problem

- Gabriyelyan, Saak, Morris, Sidney

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

Observations on the separable quotient problem for banach spaces

**Authors:**Morris, Sidney , Yost, David**Date:**2020**Type:**Text , Journal article , Article**Relation:**Axioms Vol. 9, no. 1 (2020), p.**Full Text:****Reviewed:****Description:**The longstanding Banach-Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E* onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E*. It is shown that every dual-like Banach space has an infinite-dimensional separable quotient. © 2020 by the authors.

**Authors:**Morris, Sidney , Yost, David**Date:**2020**Type:**Text , Journal article , Article**Relation:**Axioms Vol. 9, no. 1 (2020), p.**Full Text:****Reviewed:****Description:**The longstanding Banach-Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E* onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E*. It is shown that every dual-like Banach space has an infinite-dimensional separable quotient. © 2020 by the authors.

The tubby torus as a quotient group

**Authors:**Morris, Sidney**Date:**2020**Type:**Text , Journal article , Article**Relation:**Axioms Vol. 9, no. 1 (2020), p.**Full Text:****Reviewed:****Description:**Let E be any metrizable nuclear locally convex space and E the Pontryagin dual group of E. Then the topological group bE has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach-Mazur separable quotient problem. © 2020 by the author.

**Authors:**Morris, Sidney**Date:**2020**Type:**Text , Journal article , Article**Relation:**Axioms Vol. 9, no. 1 (2020), p.**Full Text:****Reviewed:****Description:**Let E be any metrizable nuclear locally convex space and E the Pontryagin dual group of E. Then the topological group bE has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach-Mazur separable quotient problem. © 2020 by the author.

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