- Avda de la Universidad de Cadiz
11519 Puerto Real, Spain (EU) - +34 956 483 346
Francisco J Garcia-Pacheco
Universidad de Cadiz, Matematicas, Faculty Member
- Kent State University, Mathematics and Computer Science, Graduate StudentTexas A&M University, Mathematics, Post-Docadd
- My field of study is Functional Analysisedit
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points,... more
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say that the topological vector space satisfies the strong Krein–Milman property. The strong Krein–Milman property trivially implies the Krein–Milman property. We provide a sufficient condition for these two properties to be equivalent in the class of Hausdorff locally convex real topological vector spaces. This sufficient condition is the Bishop–Phelps property, which we introduce for real topological vector spaces by means of uniform convergence linear topologies. We study the inheritance of the Bishop–Phelps property. Nontrivial examples of topological vector spaces failing the Krein–Milman property are also given, providing us with necessary conditions to assure that the Krein–Milman property is satisfied. Finally, a sufficient condition ...
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The index of strong rotundity is introduced. This index is used to determine how far an element of the unit sphere of a real Banach space is from being a strongly exposed point of the unit ball. This index is computed for Hilbert spaces.... more
The index of strong rotundity is introduced. This index is used to determine how far an element of the unit sphere of a real Banach space is from being a strongly exposed point of the unit ball. This index is computed for Hilbert spaces. Characterizations of the set of rotund points and the set of smooth points are provided for a better understanding of the construction of the index of strong rotundity. Finally, applications to the stereographic projection are provided.
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Exact solutions to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"><mml:mstyle scriptlevel="0" displaystyle="true"><mml:mrow><mml:munder><mml:mi>max</mml:mi><mml:mro
Abstract The supporting vectors of a matrix A are the solutions of max ∥ x ∥ 2 = 1 ∥ A x ∥ 2 2 . The generalized supporting vectors of matrices A 1 , ⋯ , A k are the solutions of max ∥ x ∥ 2 = 1 ∥ A 1 x ∥ 2 2 + ⋯ + ∥ A k x ∥ 2 2 . Notice... more
Abstract The supporting vectors of a matrix A are the solutions of max ∥ x ∥ 2 = 1 ∥ A x ∥ 2 2 . The generalized supporting vectors of matrices A 1 , ⋯ , A k are the solutions of max ∥ x ∥ 2 = 1 ∥ A 1 x ∥ 2 2 + ⋯ + ∥ A k x ∥ 2 2 . Notice that the previous optimization problem is also a boundary element problem since the maximum is attained on the unit sphere. Many problems in Physics, Statistics and Engineering can be modeled by using generalized supporting vectors. In this manuscript we first raise the generalized supporting vectors to the infinite dimensional case by solving the optimization problem max ∥ x ∥ = 1 ∑ i = 1 ∞ ∥ T i ( x ) ∥ 2 where ( T i ) i ∈ N is a sequence of bounded linear operators between Hilbert spaces H and K of any dimension. Observe that the previous optimization problem generalizes the first two. Then a unified MATLAB code is presented for computing generalized supporting vectors of a finite number of matrices. Some particular cases are considered and three novel examples are provided to which our technique applies: optimized observable magnitudes by a pure state in a quantum mechanical system, a TMS optimized coil and an optimal location problem using statistics multivariate analysis. These three examples show the wide applicability of our theoretical and computational model.
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Let [Formula: see text] be an isometric representation of a group [Formula: see text] in a Banach space [Formula: see text] over a normalizing non-discrete absolute valued division ring [Formula: see text]. If [Formula: see text] and... more
Let [Formula: see text] be an isometric representation of a group [Formula: see text] in a Banach space [Formula: see text] over a normalizing non-discrete absolute valued division ring [Formula: see text]. If [Formula: see text] and [Formula: see text] are supportive and [Formula: see text] verifies the separation property, then [Formula: see text] is 1-complemented in [Formula: see text] along [Formula: see text]. As an immediate consequence, in an isometric representation of a group in a smooth Banach space whose dual is also smooth, the subspace of [Formula: see text]-invariant vectors is [Formula: see text]-complemented.
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Let $ T:X\to Y $ be a bounded linear operator between Banach spaces $ X, Y $. A vector $ x_0\in {\mathsf{S}}_X $ in the unit sphere $ {\mathsf{S}}_X $ of $ X $ is called a supporting vector of $ T $ provided that $ \|T(x_0)\| =... more
Let $ T:X\to Y $ be a bounded linear operator between Banach spaces $ X, Y $. A vector $ x_0\in {\mathsf{S}}_X $ in the unit sphere $ {\mathsf{S}}_X $ of $ X $ is called a supporting vector of $ T $ provided that $ \|T(x_0)\| = \sup\{\|T(x)\|:\|x\| = 1\} = \|T\| $. Since matrices induce linear operators between finite-dimensional Hilbert spaces, we can consider their supporting vectors. In this manuscript, we unveil the relationship between the principal components of a matrix and its supporting vectors. Applications of our results to real-life problems are provided.
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This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that... more
This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is a space of convergence associated with any free ultrafilter of $\mathbb{N} $ N ; and that if X is not complete, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is never the space of convergence associated with any free filter of $\mathbb{N} $ N . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\ell _{\infty }(X)$ ℓ ∞ ( X ) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, th...
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We prove that if a Banach space admits a biorthogonal system whose dual part is norming, then the set of norm-attaining functionals is lineable. As a consequence, if a Banach space admits a biorthogonal system whose dual part is bounded... more
We prove that if a Banach space admits a biorthogonal system whose dual part is norming, then the set of norm-attaining functionals is lineable. As a consequence, if a Banach space admits a biorthogonal system whose dual part is bounded and its weak-star closed absolutely convex hull is a generator system, then the Banach space can be equivalently renormed so that the set of norm-attaining functionals is lineable. Finally, we prove that every infinite dimensional separable Banach space whose dual unit ball is weak-star separable has a linearly independent, countable, weak-star dense subset in its dual unit ball. As a consequence, we show the existence of linearly independent norming sets which are not the dual part of a biorthogonal system.
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A supporting vector of a matrix A for a certain norm $$\Vert \cdot \Vert $$ on $$\mathbb {R}^n$$ is a vector x such that $$\Vert x\Vert =1$$ and $$\Vert Ax\Vert =\Vert A\Vert =\displaystyle \max _{\Vert y\Vert =1}\Vert Ay\Vert $$ . In... more
A supporting vector of a matrix A for a certain norm $$\Vert \cdot \Vert $$ on $$\mathbb {R}^n$$ is a vector x such that $$\Vert x\Vert =1$$ and $$\Vert Ax\Vert =\Vert A\Vert =\displaystyle \max _{\Vert y\Vert =1}\Vert Ay\Vert $$ . In this manuscript, we characterize the existence of supporting vectors in the infinite-dimensional case for both the $$\ell _1$$ -norm and the $$\ell _{\infty }$$ -norm. Besides this characterization, our theorems provide a description of the set of supporting vectors for operators on $$\ell _{\infty }$$ and $$\ell _1$$ . As an application of our results in the finite-dimensional case for both the $$\ell _1$$ -norm and the $$\ell _{\infty }$$ -norm, we study meteorological data from stations located on the province of Cadiz (Spain). For it, we consider a matrix database with the highest temperature deviations of these stations.
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Abstract Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Very recently this concept was generalized to the one of inner points in the scope of... more
Abstract Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Very recently this concept was generalized to the one of inner points in the scope of vector spaces, which, among other things, allows to characterize the linear dimension of a vector space and also serves to provide an intrinsic characterization of linear manifolds that was not possible by using internal points. Inner points of convex sets can be seen as the affine internal points, that is, the internal points with respect to the affine hull. In this manuscript we continue this research line in the scope of topological vector spaces to study the topological structure of the inner points. First, we prove that every infinite dimensional vector space has a convex subset free of inner points which is dense in every vector topology the vector space is endowed with. Also, we find the existence of convex sets in which the set of inner points is not open in the relative topology. Following this line, we also characterize the closed convex sets for which the set of inner points is not empty and open in the relative topology. Finally, we find an example of a convex set whose set of inner points is not contained in the set of inner points of its closure.
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We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute... more
We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute semi-valued ring whose invertibles approach [Formula: see text], every neighborhood of [Formula: see text] is absorbing. We also introduce the concept of total closed unit neighborhood of zero (total closed unit) and prove that the only total closed unit of the quaternions [Formula: see text] is its closed unit ball [Formula: see text]. On the other hand, we also prove that if [Formula: see text] is an absolute semi-valued unital real algebra, then its closed unit ball [Formula: see text] is a total closed unit. Finally, we study the geometry of modules via the extreme points and the internal points, showing that no internal point can be an extreme point and that absorbance is equivalent to having [Formula: see text] as an internal point.
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Our first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if... more
Our first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. In our second result we prove that every real or complex separable normed space with dimension greater than 1 contains a balanced and absorbing subset with empty interior which is dense in the unit ball. Explicit constructions of these subsets are given.
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Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing... more
Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifolds, which was not possible by using internal points. We also characterize infinite dimensional real vector spaces by means of the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincides with the one of internal points.
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Abstract We prove that an absolute semi-valued ring is first-countable if the set of invertibles is separable and its closure contains 0. We also show that every linearly topologized topological module over an absolute semi-valued ring... more
Abstract We prove that an absolute semi-valued ring is first-countable if the set of invertibles is separable and its closure contains 0. We also show that every linearly topologized topological module over an absolute semi-valued ring whose invertibles approach 0 has the trivial topology. We also show that every sequentially compact set in a topological module is bounded if the module is over an absolute semi-valued ring whose set of invertibles is separable and its closure contains 0. Finally, we find sufficient conditions for a sequentially compact neighborhood of 0 to force the corresponding module to be finitely generated.
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ABSTRACT In this brief communication we propose a vector-valued version of Lorentz’ intrinsic characterization of almost convergence, for which we find a legitimate extension of the concept of Banach limit to vector-valued sequences.... more
ABSTRACT In this brief communication we propose a vector-valued version of Lorentz’ intrinsic characterization of almost convergence, for which we find a legitimate extension of the concept of Banach limit to vector-valued sequences. Banach spaces 1-complemented in their biduals admit vector-valued Banach limits, whereas c 0 does not.
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Self-adjoint operators in smooth Banach spaces have been already defined in recent works. Here, we extend the concept of adjoint of an operator to the scope of (non-necessarily Hilbert) Banach spaces, obtaining in particular the notion of... more
Self-adjoint operators in smooth Banach spaces have been already defined in recent works. Here, we extend the concept of adjoint of an operator to the scope of (non-necessarily Hilbert) Banach spaces, obtaining in particular the notion of self-adjoint operator in the non-smooth case. As a consequence, we define the probability density operator on Banach spaces and verify most of its well-known properties.
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Given a ring endowed with a ring order, we provide sufficient conditions for the order topology induced by the ring order to become a ring topology (analogous results for module orders are consequently derived). Finally, the notions of... more
Given a ring endowed with a ring order, we provide sufficient conditions for the order topology induced by the ring order to become a ring topology (analogous results for module orders are consequently derived). Finally, the notions of Radon and regular measures are transported to the scope of module-valued measures through module orders. Classical characterizations of these measures are obtained as well as the hereditariness of regularity for conditional ring-valued measures.