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Research paper thumbnail of Inequivalent representations of the dual space

Revista de la Unión Matemática Argentina

We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ... more We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ] for p ∈ [1, ∞). We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if A is a proper closed subspace of C[0, 1], there always exists a closed subspace A ⊥ (with the same definition as for L 2 [0, 1]) such that A ∩ A ⊥ = {0} and A ⊕ A ⊥ = C[0, 1]. Thus, the geometry of C[0, 1] can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator A on C[0, 1] has a uniquely defined adjoint A * defined on C[0, 1], and both can be extended to bounded linear operators on L 2 [0, 1]. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to Lp[R n ]. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on C[0, 1] is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.

Research paper thumbnail of The Common Vision Project: Four Reactions

Notices of the American Mathematical Society, 2017

The Common Vision project seeks to modernize undergraduate programs in the mathematical sciences,... more The Common Vision project seeks to modernize undergraduate programs in the mathematical sciences, especially courses taken in the first two years. It is a joint effort of the American Mathematical Association of Two-Year Colleges, the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. Their report 1 reflects a synthesis of common themes in seven curricular guides published by these five associations along with more recent research and input. The report calls on the community to: (1) update curricula, (2) articulate clear pathways between curricula driven by changes at the K-12 level and the first courses students take in college, (3) scale up the use of evidence-based pedagogical methods, (4) find ways to remove barriers facing students at critical transition points (e.g., placement, transfer), and (5) establish stronger connections with other disciplines. It urges institutions to provide faculty with training, resources, and rewards for their efforts to meet these goals. The Notices asked four mathematicians to write short pieces summarizing their reactions to the Common Vision report. These pieces appear below.

Research paper thumbnail of Spectral characterization for von Neumann’s iterative algorithm in ℝ n

Involve, a Journal of Mathematics, 2013

Research paper thumbnail of Complex extreme measurable selections

Journal of the Australian Mathematical Society, 1995

We give a characterization of complex extreme measurable selections for a suitable set-valued map... more We give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

Research paper thumbnail of A Grothendieck compactness principle for the Mackey dual topology

Journal of Mathematical Analysis and Applications, 2014

Research paper thumbnail of Complex Strongly Extreme Points in Quasi-Normed Spaces

Journal of Mathematical Analysis and Applications, 1996

We study the complex strongly extreme points of bounded subsets of continuously quasi-normed vect... more We study the complex strongly extreme points of bounded subsets of continuously quasi-normed vector spaces X over ‫.ރ‬ When X is a complex normed linear Ž. space, these points are the complex analogues of the familiar real strongly extreme points. We show that if X is a complex Banach space then the complex strongly extreme points of B admit several equivalent formulations some of which X are in terms of ''pointwise'' versions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of B and B where 0p-ϱ, X and each X , j g I, are complex l Ž X. L Ž , X. j p j jgI p Banach spaces.

Research paper thumbnail of On the difficulty of preserving monotonicity via projections and related results

A subspace V of a Banach space X is said to be complemented if there exists a (bounded) projectio... more A subspace V of a Banach space X is said to be complemented if there exists a (bounded) projection mapping X onto V. Obviously all subspaces of finitedimension are complemented. The goal of this note is to show that there are (relatively) few monotonically complemented subspaces of finite-dimension in X =(C[a, b], Y Y∞); that is, finite-dimensional subspaces V ⊂ X for which there exists a projection P: X → V such that Pf is monotone-increasing whenever f is. We obtain several corollaries from this consideration, including a result describing the difficulty of preserving n-convexity via a projection. © 2010 Universidad de Jaén

Research paper thumbnail of A Note on the Existence of Shape-Preserving Projections

Rocky Mountain Journal of Mathematics, 2007

Research paper thumbnail of A characterization and equations for minimal shape-preserving projections

Journal of Approximation Theory, 2006

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B = B(X, V) the ... more Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B = B(X, V) the space of all bounded linear operators from X into V; let P(X, V) be the set of all projections in B. For a given cone S ⊂ X, we denote by P = P S (X, V) the set of operators P ∈ P such that P S ⊂ S. When P S = ∅, we characterize those P ∈ P S for which P is minimal. This characterization is then utilized in several applications and examples.

Research paper thumbnail of Complex convexity in Lebesgue-Bochner Function Spaces

Transactions of the American Mathematical Society, 1996

Complex geometric properties of continuously quasi-normed spaces are introduced and their relatio... more Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space X X to L p ( μ , X ) L^p(\mu , X) , for 0 > p > ∞ 0 > p > \infty . Local versions of these properties and results are also considered.

Research paper thumbnail of Inequivalent representations of the dual space

Revista de la Unión Matemática Argentina

We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ... more We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ] for p ∈ [1, ∞). We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if A is a proper closed subspace of C[0, 1], there always exists a closed subspace A ⊥ (with the same definition as for L 2 [0, 1]) such that A ∩ A ⊥ = {0} and A ⊕ A ⊥ = C[0, 1]. Thus, the geometry of C[0, 1] can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator A on C[0, 1] has a uniquely defined adjoint A * defined on C[0, 1], and both can be extended to bounded linear operators on L 2 [0, 1]. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to Lp[R n ]. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on C[0, 1] is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.

Research paper thumbnail of Inequivalent representations of the dual space

Revista de la Unión Matemática Argentina

We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ... more We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ] for p ∈ [1, ∞). We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if A is a proper closed subspace of C[0, 1], there always exists a closed subspace A ⊥ (with the same definition as for L 2 [0, 1]) such that A ∩ A ⊥ = {0} and A ⊕ A ⊥ = C[0, 1]. Thus, the geometry of C[0, 1] can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator A on C[0, 1] has a uniquely defined adjoint A * defined on C[0, 1], and both can be extended to bounded linear operators on L 2 [0, 1]. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to Lp[R n ]. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on C[0, 1] is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.

Research paper thumbnail of The Common Vision Project: Four Reactions

Notices of the American Mathematical Society, 2017

The Common Vision project seeks to modernize undergraduate programs in the mathematical sciences,... more The Common Vision project seeks to modernize undergraduate programs in the mathematical sciences, especially courses taken in the first two years. It is a joint effort of the American Mathematical Association of Two-Year Colleges, the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. Their report 1 reflects a synthesis of common themes in seven curricular guides published by these five associations along with more recent research and input. The report calls on the community to: (1) update curricula, (2) articulate clear pathways between curricula driven by changes at the K-12 level and the first courses students take in college, (3) scale up the use of evidence-based pedagogical methods, (4) find ways to remove barriers facing students at critical transition points (e.g., placement, transfer), and (5) establish stronger connections with other disciplines. It urges institutions to provide faculty with training, resources, and rewards for their efforts to meet these goals. The Notices asked four mathematicians to write short pieces summarizing their reactions to the Common Vision report. These pieces appear below.

Research paper thumbnail of Spectral characterization for von Neumann’s iterative algorithm in ℝ n

Involve, a Journal of Mathematics, 2013

Research paper thumbnail of Complex extreme measurable selections

Journal of the Australian Mathematical Society, 1995

We give a characterization of complex extreme measurable selections for a suitable set-valued map... more We give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

Research paper thumbnail of A Grothendieck compactness principle for the Mackey dual topology

Journal of Mathematical Analysis and Applications, 2014

Research paper thumbnail of Complex Strongly Extreme Points in Quasi-Normed Spaces

Journal of Mathematical Analysis and Applications, 1996

We study the complex strongly extreme points of bounded subsets of continuously quasi-normed vect... more We study the complex strongly extreme points of bounded subsets of continuously quasi-normed vector spaces X over ‫.ރ‬ When X is a complex normed linear Ž. space, these points are the complex analogues of the familiar real strongly extreme points. We show that if X is a complex Banach space then the complex strongly extreme points of B admit several equivalent formulations some of which X are in terms of ''pointwise'' versions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of B and B where 0p-ϱ, X and each X , j g I, are complex l Ž X. L Ž , X. j p j jgI p Banach spaces.

Research paper thumbnail of On the difficulty of preserving monotonicity via projections and related results

A subspace V of a Banach space X is said to be complemented if there exists a (bounded) projectio... more A subspace V of a Banach space X is said to be complemented if there exists a (bounded) projection mapping X onto V. Obviously all subspaces of finitedimension are complemented. The goal of this note is to show that there are (relatively) few monotonically complemented subspaces of finite-dimension in X =(C[a, b], Y Y∞); that is, finite-dimensional subspaces V ⊂ X for which there exists a projection P: X → V such that Pf is monotone-increasing whenever f is. We obtain several corollaries from this consideration, including a result describing the difficulty of preserving n-convexity via a projection. © 2010 Universidad de Jaén

Research paper thumbnail of A Note on the Existence of Shape-Preserving Projections

Rocky Mountain Journal of Mathematics, 2007

Research paper thumbnail of A characterization and equations for minimal shape-preserving projections

Journal of Approximation Theory, 2006

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B = B(X, V) the ... more Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B = B(X, V) the space of all bounded linear operators from X into V; let P(X, V) be the set of all projections in B. For a given cone S ⊂ X, we denote by P = P S (X, V) the set of operators P ∈ P such that P S ⊂ S. When P S = ∅, we characterize those P ∈ P S for which P is minimal. This characterization is then utilized in several applications and examples.

Research paper thumbnail of Complex convexity in Lebesgue-Bochner Function Spaces

Transactions of the American Mathematical Society, 1996

Complex geometric properties of continuously quasi-normed spaces are introduced and their relatio... more Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space X X to L p ( μ , X ) L^p(\mu , X) , for 0 > p > ∞ 0 > p > \infty . Local versions of these properties and results are also considered.

Research paper thumbnail of Inequivalent representations of the dual space

Revista de la Unión Matemática Argentina

We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ... more We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[R n ] for p ∈ [1, ∞). We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if A is a proper closed subspace of C[0, 1], there always exists a closed subspace A ⊥ (with the same definition as for L 2 [0, 1]) such that A ∩ A ⊥ = {0} and A ⊕ A ⊥ = C[0, 1]. Thus, the geometry of C[0, 1] can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator A on C[0, 1] has a uniquely defined adjoint A * defined on C[0, 1], and both can be extended to bounded linear operators on L 2 [0, 1]. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to Lp[R n ]. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on C[0, 1] is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.