## 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...

## 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

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...

## 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...

##
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...

## Linear Operators in Banach Spaces

in** Spectral Theory and Differential Operators**

May 2018; p ublished online September 2018 .

Chapter. Subjects: Pure Mathematics. 30346 words.

Three main themes run through this chapter: compact linear operators, measures of non-compactness, and Fredholm and semi-Fredholm maps. Connections are established between these themes so...

## Smooth functions and partitions of unity on certain Banach spaces

in** The Quarterly Journal of Mathematics**

December 1996 .

Journal Article. Subjects: Pure Mathematics. 0 words.

## On the holomorphic rigidity of linear operators on complex Banach spaces

in** The Quarterly Journal of Mathematics**

September 1999; p ublished online September 1999 .

Journal Article. Subjects: Pure Mathematics. 0 words.

## Convex‐Transitive Banach Spaces, Big Points, and the Duality Mapping

in** The Quarterly Journal of Mathematics**

September 2002; p ublished online September 2002 .

Journal Article. Subjects: Pure Mathematics. 0 words.

## SMOOTH FUNCTIONS AND PARTITIONS OF UNITY ON CERTAIN BANACH SPACES

in** The Quarterly Journal of Mathematics**

December 1996 .

Journal Article. Subjects: Pure Mathematics. 0 words.

## A KREIN-ŠMULIAN TYPE RESULT IN BANACH SPACES

in** The Quarterly Journal of Mathematics**

June 1997; p ublished online June 1997 .

Journal Article. Subjects: Pure Mathematics. 0 words.