scholarly journals A Monge-Ampère norm for delta-plurisubharmonic functions

2005 ◽  
Vol 97 (2) ◽  
pp. 201 ◽  
Author(s):  
Urban Cegrell ◽  
Jonas Wiklund
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Total Variation ◽  
Dual Space ◽  
Energy Class ◽  
Plurisubharmonic Functions ◽  
Generalized Complex ◽  
Monge Ampere

We consider differences of plurisubharmonic functions in the energy class $\mathcal{F}$ as a linear space, and equip this space with a norm, depending on the generalized complex Monge-Ampère operator, turning the linear space into a Banach space $\delta \mathcal{F}$. Fundamental topological questions for this space is studied, and we prove that $\delta\mathcal{F}$ is not separable. Moreover we investigate the dual space. The study is concluded with comparison between $\delta \mathcal{F}$ and the space of delta-plurisubharmonic functions, with norm depending on the total variation of the Laplace mass, studied by the first author in an earlier paper [7].

C(X) As A Dual Space

1972 ◽  
Vol 24 (3) ◽  
pp. 485-491 ◽  
Author(s):  
E. G. Manes
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Dual Space ◽  
Minkowski Functional ◽  
Topological Linear Space ◽  
Open Set ◽  
Topological Linear

It is known [1] that for compact Hausdorff X, C(X) is the dual of a Banach space if and only if X is hyperstonian, that is the closure of an open set in X is again open and the carriers of normal measures in C(X)* have dense union in X. With the desiratum of proving that C(X) is always the dual of some sort of space we broaden the concept of Banach space as follows. A Banach space may be comfortably regarded as a pair (E, B) where E is a topological linear space and B is a subset of E ; the requisite property is that the Minkowski functional of B be a complete norm whose topology coincides with that of E.

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals On the modulus ofU-convexity

2005 ◽  
Vol 2005 (1) ◽  
pp. 59-66 ◽  
Author(s):  
Satit Saejung
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Normal Structure ◽  
Uniform Normal Structure

We prove that the moduli ofU-convexity, introduced by Gao (1995), of the ultrapowerX˜of a Banach spaceXand ofXitself coincide wheneverXis super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove thatuX(1)>0implies that bothXand the dual spaceX∗ofXhave uniform normal structure and hence the “worth” property in Corollary 7 of Mazcuñán-Navarro (2003) can be discarded.

scholarly journals The space $D$ in several variables: random variables and higher moments

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Random Variables ◽  
Higher Moments ◽  
Multilinear Form ◽  
Standard Case ◽  
Random Functions ◽  
Several Variables ◽  
Random Elements ◽  
Point Evaluations

We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a description of the dual space, and we show that a bounded multilinear form always is measurable with respect to the $\sigma$-field generated by the point evaluations. These results are used to study random functions in the space. (I.e., random elements of the space.) In particular, we give results on existence of moments (in different senses) of such random functions, and we give an application to the Zolotarev distance between two such random functions.

scholarly journals Complex interpolation of a Banach space with its dual

2000 ◽  
Vol 87 (2) ◽  
pp. 200
Author(s):  
Frédérique Watbled
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Vector Space ◽  
Unconditional Basis ◽  
Scalar Product ◽  
Dual Space ◽  
Function Spaces ◽  
Complex Interpolation

Let $X$ be a Banach space compatible with its antidual $\overline{X^*}$, where $\overline{X^*}$ stands for the vector space $X^*$ where the multiplication by a scalar is replaced by the multiplication $\lambda \odot x^* = \overline{\lambda} x^*$. Let $H$ be a Hilbert space intermediate between $X$ and $\overline{X^*}$ with a scalar product compatible with the duality $(X,X^*)$, and such that $X \cap \overline{X^*}$ is dense in $H$. Let $F$ denote the closure of $X \cap \overline{X^*}$ in $\overline{X^*}$ and suppose $X \cap \overline{X^*}$ is dense in $X$. Let $K$ denote the natural map which sends $H$ into the dual of $X \cap F$ and for every Banach space $A$ which contains $X \cap F$ densely let $A'$ be the realization of the dual space of $A$ inside the dual of $X \cap F$. We show that if $\vert \langle K^{-1}a, K^{-1}b \rangle_H \vert \leq \parallel a \parallel_{X'} \parallel b \parallel_{F'}$ whenever $a$ and $b$ are both in $X' \cap F'$ then $(X, \overline{X^*})_{\frac12} = H$ with equality of norms. In particular this equality holds true if $X$ embeds in $H$ or $H$ embeds densely in $X$. As other particular cases we mention spaces $X$ with a $1$-unconditional basis and Köthe function spaces on $\Omega$ intermediate between $L^1(\Omega)$ and $L^\infty(\Omega)$.

scholarly journals On a Characterisation of Differentiability of the Norm of a Normed Linear Space

1971 ◽  
Vol 12 (1) ◽  
pp. 106-114 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Dual Norm ◽  
Uniform Rotundity ◽  
Strong Differentiability ◽  
Differentiability Properties

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

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