## Banach space

Overview page. Subjects: Mathematics.

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

## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## Elements of Geometry of Balls in Banach Spaces

September 2018; p ublished online April 2019 .

Book. Subjects: Pure Mathematics. 176 pages.

One of the subjects of functional analysis is classification of Banach spaces depending on various properties of the unit ball. The need of such considerations comes from a number of...

## 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....

## 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...

## 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...

##
On Emerton’s *p*-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...

## 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...

##
*C*
^{1}-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...

## 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...

## 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...

##
SMOOTH NORMS AND APPROXIMATION IN BANACH SPACES OF THE TYPE 𝒞(*K*)

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...