scholarly journals A Reverse Theorem on the·-w*Continuity of the Dual Map

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
Mienie de Kock ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Equivalent Norm ◽  
Infinite Dimensional ◽  
Riesz Property ◽  
Dual Map

Given a Banach spaceX,x∈𝖲X, and𝖩Xx=x*∈𝖲X*:x*x=1, we define the set𝖩X*xof allx*∈𝖲X*for which there exist two sequencesxnn∈N⊆𝖲X∖{x}andxn*n∈N⊆𝖲X*such thatxnn∈Nconverges tox,xn*n∈Nhas a subnetw*-convergent tox*, andxn*xn=1for alln∈N. We prove that ifXis separable and reflexive andX*enjoys the Radon-Riesz property, then𝖩X*xis contained in the boundary of𝖩Xxrelative to𝖲X*. We also show that ifXis infinite dimensional and separable, then there exists an equivalent norm onXsuch that the interior of𝖩Xxrelative to𝖲X*is contained in𝖩X*x.

scholarly journals The Fixed Point Property inc0with an Equivalent Norm

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Berta Gamboa de Buen ◽  
Fernando Núñez-Medina
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Dual Space ◽  
Equivalent Norm ◽  
Dimensional Subspace ◽  
Fixed Point Property ◽  
Point Property ◽  
Infinite Dimensional ◽  
Weak Fixed Point Property ◽  
Infinite Dimensional Subspace

We study the fixed point property (FPP) in the Banach spacec0with the equivalent norm‖⋅‖D. The spacec0with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of(c0,‖⋅‖D)contains a complemented asymptotically isometric copy ofc0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of(c0,‖⋅‖D)which are notω-compact and do not contain asymptotically isometricc0—summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space(c0,‖⋅‖D),and we give some of its properties. We also prove that the dual space of(c0,‖⋅‖D)over the reals is the Bynum spacel1∞and that every infinite-dimensional subspace ofl1∞does not have the fixed point property.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

scholarly journals Characteristic Functions and Borel Exceptional Values ofE-Valued Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Zhaojun Wu ◽  
Zuxing Xuan
Keyword(s):  
Banach Space ◽  
Meromorphic Functions ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Characteristic Functions ◽  
Infinite Dimensional ◽  
Borel Exceptional Values

The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values ofE-valued meromorphic functions from theℂR={z:|z|<R},  0<R≤+∞to an infinite-dimensional complex Banach spaceEwith a Schauder basis. Results obtained extend the relative results by Xuan, Wu and Yang, Bhoosnurmath, and Pujari.

Holomorphic functions with distinguished properties on infinite dimensional spaces

2018 ◽  
Vol 70 (3) ◽  
pp. 797-811
Author(s):  
Thiago R Alves ◽  
Geraldo Botelho
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Algebraic Structures ◽  
Infinite Dimensional ◽  
Zero Function ◽  
Bounded Holomorphic

Abstract In this paper, we develop a method to construct holomorphic functions that exist only on infinite dimensional spaces. The following types of holomorphic functions f:U→ℂ on some open subsets U of an infinite dimensional complex Banach space are constructed: (1) f is bounded holomorphic on U and is continuously, but not uniformly continuously extended to U¯; (2) f is continuous on U¯ and holomorphic of bounded type on U, but f is unbounded on U; (3) f is holomorphic of bounded type on U and f cannot be continuously extended to U¯. The technique we develop is powerful enough to provide, in the cases (2) and (3) above, large algebraic structures formed by such functions (up to the zero function, of course).

scholarly journals On the M-hypercyclicity of cosine function on Banach spaces

2019 ◽  
Vol 38 (3) ◽  
pp. 133-140
Author(s):  
Abdelaziz Tajmouati ◽  
Abdeslam El Bakkali ◽  
Ahmed Toukmati
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Infinite Dimensional ◽  
Uniformly Continuous

In this paper we introduce and study the M-hypercyclicity of strongly continuous cosine function on separable complex Banach space, and we give the criteria for cosine function to be M-hypercyclic. We also prove that every separable infinite dimensional complex Banach space admits a uniformly continuous cosine function.

scholarly journals Quasi-regularity in Optimization

1986 ◽  
Vol 41 (2) ◽  
pp. 188-192 ◽  
Author(s):  
Kung-Fu Ng ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Lagrange Multiplier ◽  
Optimization Problems ◽  
General Situation ◽  
Lagrange Multiplier Rule ◽  
Infinite Dimensional ◽  
Space Setting ◽  
Multiplier Rule ◽  
Dimensional Version ◽  
Definition Of

AbstractThe notion of quasi-regularity, defined for optimization problems in Rn, is extended to the Banach space setting. Examples are given to show that our definition of quasi-regularity is more natural than several other possibilities in the general situation. An infinite dimensional version of the Lagrange multiplier rule is established.

scholarly journals Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 133
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Main Question ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Taylor Polynomials

In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.

A Banach Space which is Fully 2-Rotund but not Locally Uniformly Rotund

1983 ◽  
Vol 26 (1) ◽  
pp. 118-120 ◽  
Author(s):  
T. Polak ◽  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Equivalent Norm ◽  
Image Position

AbstractA Banach space is fully 2-rotund if (xn) converges whenever ‖xn + xm‖ converges as m, n → ∞ and locally uniformly rotund if xn → x whenever ‖xn‖ and ‖(xn + x)/2‖ → ‖x‖.We show that I2 with the equivalent normis fully 2-rotund but not locally uniformly rotund, thus answering in the negative a question first raised by Fan and Glicksberg in 1958.

Biholomorphic Mappings on Banach Spaces

2019 ◽  
Vol 62 (4) ◽  
pp. 913-924
Author(s):  
H. Carrión ◽  
P. Galindo ◽  
M. L. Lourenço
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Open Subset ◽  
Separable Hilbert Space ◽  
Derivative Operator ◽  
Infinite Dimensional ◽  
Dimensional Version ◽  
Dual Banach Space ◽  
Bounded Convex ◽  
Power Bounded

AbstractWe present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.

On James' Quasi-Reflexive Banach Space as a Banach Algebra

1980 ◽  
Vol 32 (5) ◽  
pp. 1080-1101 ◽  
Author(s):  
Alfred D. Andrew ◽  
William L. Green
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Reflexive Banach Space ◽  
Equivalent Norm ◽  
New Techniques ◽  
Codimension One ◽  
Gelfand Transform ◽  
Schauder Bases ◽  
Second Dual ◽  
Arens Multiplication

In [4] and [5], R. C. James introduced a non-reflexive Banach space J which is isometric to its second dual. Developing new techniques in the theory of Schauder bases, James identified J**, showed that the canonical image of J in J** is of codimension one, and proved that J** is isometric to J.In Section 2 of this paper we show that J, equipped with an equivalent norm, is a semi-simple (commutative) Banach algebra under point wise multiplication, and we determine its closed ideals. We use the Arens multiplication and the Gelfand transform to identify J**, which is in fact just the algebra obtained from J by adjoining an identity.

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