scholarly journals Asymptotic-norming and Mazur intersection properties in Bochner function spaces

1993 ◽  
Vol 48 (2) ◽  
pp. 177-186 ◽  
Author(s):  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Function Spaces ◽  
Intersection Property ◽  
Asplund Space

A Banach space X has the asymptotic-norming property if and only if the Lebesgue-Bochner function space Lp (μ, X) has the asumptotic-norming property for p with 1 < p < ∞. It follows that a Banach space X is Hahn-Banach smooth if and only if Lp (μ, X) is Hahn-Banach smooth for p with 1 < p < ∞. We also show that for p with 1 < p < ∞, (1) if X has the compact Mazur intersection property then so does Lp(μ, X); (2) if the measure μ is not purely atomic, then the space Lp(μ, X) has the Mazur intersection property if and only if X is an Asplund space and has the Mazur intersection property.

scholarly journals Balls intersection properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Necessary And Sufficient Conditions ◽  
Asplund Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

scholarly journals P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type

2017 ◽  
Vol 3 (1) ◽  
pp. 221
Author(s):  
Yulia Romadiastri
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Orlicz Function ◽  
Function Spaces ◽  
Convexity Property

<div style="text-align: justify;">In this paper, we described about Musielak-Orlicz function spaces of Bochner type. It has been obtained that Musielak-Orlicz function space <a href="https://www.codecogs.com/eqnedit.php?latex=L_\phi(\mu,X)" target="_blank"><img title="L_\phi(\mu,X)" src="https://latex.codecogs.com/gif.latex?L_\phi(\mu,X)" alt="" /></a> of Bochner type becomes a Banach space. It is described also about P-convexity of Musielak-Orlicz function space <a href="https://www.codecogs.com/eqnedit.php?latex=\small&amp;space;L_\phi(\mu,X)" target="_blank"><img title="\small L_\phi(\mu,X)" src="https://latex.codecogs.com/gif.latex?\small&amp;space;L_\phi(\mu,X)" alt="" /></a> of Bochner type. It is proved that the Musielak-Orlicz function space <a href="https://www.codecogs.com/eqnedit.php?latex=\small&amp;space;L_\phi(\mu,X)" target="_blank"><img title="\small L_\phi(\mu,X)" src="https://latex.codecogs.com/gif.latex?\small&amp;space;L_\phi(\mu,X)" alt="" /></a> of Bochner type is P-convex if and only if both spaces <a href="https://www.codecogs.com/eqnedit.php?latex=\small&amp;space;L_\phi" target="_blank"><img title="\small L_\phi" src="https://latex.codecogs.com/gif.latex?\small&amp;space;L_\phi" alt="" /></a> and X are P-convex.©2017 JNSMR UIN Walisongo. All rights reserved.</div>

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

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 POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

A Stability Property of a Class of Banach Spaces Not Containing c0

1992 ◽  
Vol 35 (1) ◽  
pp. 56-60 ◽  
Author(s):  
Patrick N. Dowling
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Function Space ◽  
Stability Property ◽  
Banach Function Space ◽  
Ideal Space

AbstractLet E be a Banach ideal space and X be a Banach space. The Banach function space E(X) does not contain a copy of C0 if and only if neither E nor X contains a copy of c0. Some extensions of this result are also noted.

scholarly journals Duals of vector-valued Köthe function spaces

1992 ◽  
Vol 112 (1) ◽  
pp. 165-174 ◽  
Author(s):  
Miguel Florencio ◽  
Pedro J. Paúl ◽  
Carmen Sáez
Keyword(s):  
Function Space ◽  
Normed Space ◽  
Function Spaces ◽  
Integrable Functions ◽  
Nikodym Property ◽  
Köthe Dual ◽  
Vector Valued

AbstractLet Λ be a perfect Köthe function space in the sense of Dieudonné, and Λ× its Köthe-dual. Let E be a normed space. Then the topological dual of the space Λ(E) of Λ-Bochner integrable functions equals the corresponding Λ×(E′) if and only if E′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind.

scholarly journals Banach Lattice Structures and Concavifications in Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 127
Author(s):  
Lucia Agud ◽  
Jose Manuel Calabuig ◽  
Maria Aranzazu Juan ◽  
Enrique A. Sánchez Pérez
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Function Space ◽  
Lattice Structure ◽  
Banach Function Space ◽  
Finite Measure ◽  
Multiplication Operators ◽  
Spaces Of Operators ◽  
Finite Measure Space ◽  
Local Properties

Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

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