##
Banach space

Overview page. Subjects: Mathematics.

A complete normed vector space on the real or complex numbers.

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Lipschitz Homogeneous Banach Spaces

in** The Quarterly Journal of Mathematics**

December 2003; p ublished online December 2003 .

Journal Article. Subjects: Pure Mathematics. 0 words.

Define a Lipschitz homogeneous space to be a Banach space Lipschitz equivalent to all its closed subspaces. It is proved that a Lipschitz homogeneous Banach space with the Radon–Nikodym...

##
On Weakly‐Compactly Generated Asplund Banach spaces

in** The Quarterly Journal of Mathematics**

March 2004; p ublished online March 2004 .

Journal Article. Subjects: Pure Mathematics. 0 words.

In this paper we give a sufficient and necessary condition for a Banach space to be weakly‐compactly generated and Asplund. This condition links the weak^{*} and the norm topologies of its...

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POSITIVELY NORMING SETS IN BANACH FUNCTION SPACES

in** The Quarterly Journal of Mathematics**

September 2014; p ublished online November 2013 .

Journal Article. Subjects: Pure Mathematics. 0 words.

The notion of positively norming set, a specific definition of norming type sets for Banach lattices, is analyzed. We show that the size of positively norming sets (in terms of compactness...

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High Order Smoothness and Roughness in Banach Spaces

in** The Quarterly Journal of Mathematics**

March 2000; p ublished online March 2000 .

Journal Article. Subjects: Pure Mathematics. 0 words.

We study the existence of equivalent norms nowhere *k*-times Fréchet differentiable (*k* ≥ 2) on Banach spaces, introducing the notion of high order roughness. We also study the existence of...

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CONVERGENCE OF MONOMIAL EXPANSIONS IN BANACH SPACES

in** The Quarterly Journal of Mathematics**

September 2012; p ublished online January 2011 .

Journal Article. Subjects: Pure Mathematics. 0 words.

If *E* is a Banach sequence space, then each holomorphic function defines a formal power series ∑_{α}
*c*
_{α}(*f*) *z*
^{α}. The problem of when such an expansion...

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SMOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES

in** The Quarterly Journal of Mathematics**

September 2014; p ublished online December 2013 .

Journal Article. Subjects: Pure Mathematics. 0 words.

Let *X* be a separable real Banach space having a *k*-times continuously Fréchet differentiable (i.e. *C*
^{
k
}-smooth) norm where *k*∈{1, …, ∞}. We show that...

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Vector Integration and Stochastic Integration in Banach Spaces

in** The Oxford Handbook of Functional Data Analysis**

November 2010; p ublished online August 2018 .

Article. Subjects: Mathematics. 13039 words.

This article deals with vector integration and stochastic integration in Banach spaces. In particular, it considers the theory of integration with respect to vector measures with finite...

##
Banach space

Overview page. Subjects: Mathematics.

A complete normed vector space on the real or complex numbers.

##
Banach space

in** The Concise Oxford Dictionary of Mathematics**

January 2009; p ublished online January 2009 .

Reference Entry. Subjects: Pure Mathematics. 13 words.

A complete normed vector space on the real or complex numbers.

##
Banach space

in** The Concise Oxford Dictionary of Mathematics**

January 2014; p ublished online September 2014 .

Reference Entry. Subjects: Pure Mathematics. 13 words.

A complete normed vector space on the real or complex numbers.

##
WEAKLY METRIZABILITY OF SPHERES AND RENORMINGS OF BANACH SPACES

in** The Quarterly Journal of Mathematics**

March 2016; p ublished online January 2016 .

Journal Article. Subjects: Pure Mathematics. 5777 words.

We study the metrizability of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and its relationships with other geometrical properties of norms....

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NUMERICAL RADIUS OF RANK-1 OPERATORS ON BANACH SPACES

in** The Quarterly Journal of Mathematics**

March 2014; p ublished online January 2013 .

Journal Article. Subjects: Pure Mathematics. 0 words.

We study the rank-1 numerical index of a Banach space, namely the infimum of the numerical radii of those rank-1 operators on the space which have norm 1. We show that the rank-1 numerical...

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Lyapunov stability of abstract nonlinear dynamic system in Banach space

in** IMA Journal of Mathematical Control and Information**

March 2003; p ublished online March 2003 .

Journal Article. Subjects: Mathematics. 0 words.

The Lyapunov stability theory for nonlinear time‐varying dynamic system in Banach space is given in this paper. The Lyapunov stable theorem and the Barbashin–Krasovskii–LaSalle invariant set...

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On Emerton’s <i>p</i>-adic Banach Spaces

in** International Mathematics Research Notices**

January 2010; p ublished online February 2010 .

Journal Article. Subjects: Mathematics. 9547 words.

Given a linear algebraic group *G* defined over a number field, Emerton [10] has defined a sequence of *p*-adic Banach Spaces . These spaces are representations of the group of adele points of *G*null...

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Spin spaces and positive decomposition of linear maps on ordered Banach spaces

in** The Quarterly Journal of Mathematics**

March 2005; p ublished online March 2005 .

Journal Article. Subjects: Pure Mathematics. 0 words.

Spin factors and generalizations are used to revisit positive generation of *B*(*E*, *F*), where *E* and *F* are ordered Banach spaces. Interior points of *B*(*E*, *F*)_{+} are discussed and in many cases it...

##
<i>C</i>
<sup>1</sup>-fine approximation of functions on Banach spaces with unconditional basis

in** The Quarterly Journal of Mathematics**

March 2005; p ublished online March 2005 .

Journal Article. Subjects: Pure Mathematics. 0 words.

We show that if *X* is a Banach space having an unconditional basis and a *C*
^{
p
}-smooth Lipschitz bump function, then for every *C*
^{1}-smooth...

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Convergence of Newton's method and uniqueness of the solution of equations in Banach space

in** IMA Journal of Numerical Analysis**

January 2000; p ublished online January 2000 .

Journal Article. Subjects: Mathematics. 0 words.

Under the hypothesis that the derivative satisfies some kind of weak Lipschitz condition, exact estimates of the radii of convergence ball of Newton's method and of the uniqueness ball for...

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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

February 2012; p ublished online October 2017 .

Book. Subjects: Mathematics. 440 pages.

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because...

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SMOOTH NORMS AND APPROXIMATION IN BANACH SPACES OF THE TYPE 𝒞(<i>K</i>)

in** The Quarterly Journal of Mathematics**

June 2007; p ublished online March 2007 .

Journal Article. Subjects: Pure Mathematics. 0 words.

**Abstract**

Two results are proved about the Banach space *X* = 𝒞(*K*), where *K* is compact and Hausdorff. The first concerns smooth approximation: let *m* be a...

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Criteria for exponential stability of linear differential equations with positive evolution on ordered Banach spaces

in** IMA Journal of Mathematical Control and Information**

September 2010; p ublished online July 2010 .

Journal Article. Subjects: Mathematics. 0 words.

In this paper, the problem of exponential stability of the zero state equilibrium of a linear differential equation defined by an operator-valued function with positive evolution on an...