scholarly journals On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

2003 ◽  
Vol 155 (2) ◽  
pp. 171-182 ◽  
Author(s):  
Artur Michalak
Keyword(s):  
Banach Space ◽  
Unit Interval ◽  
Monotonic Functions ◽  
Points Of Discontinuity

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

scholarly journals Best Simultaneous approximation of quasi-continuous functions by monotone functions

1991 ◽  
Vol 50 (3) ◽  
pp. 391-408 ◽  
Author(s):  
Salem M. A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Continuous Functions ◽  
Unit Interval ◽  
Closed Convex Cone ◽  
Monotone Functions ◽  
Measurable Functions ◽  
Best Simultaneous Approximation ◽  
Nondecreasing Functions

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

scholarly journals Biconcave-function characterisations of UMD and Hilbert spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Jinsik Mok Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Banach Space ◽  
Unit Interval ◽  
Martingale Differences ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Image Position ◽  
Umd Banach Spaces

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only iffor all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 andIf the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

An example regarding Kalton's paper ‘isomorphisms between spaces of vector-valued continuous functions’

2021 ◽  
pp. 1-5
Author(s):  
Félix Cabello Sánchez
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Continuous Functions ◽  
Unit Interval ◽  
Finite Covering ◽  
Covering Dimension ◽  
Striking Result ◽  
Vector Valued ◽  
Locally Convex

Abstract The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_{0}$ which is isomorphic to a closed subspace of a space with a basis and $C(I,\,X)$ is linearly homeomorphic to $C(\Delta ,\, X)$ , then $X$ is locally convex, i.e., a Banach space. We will show that Kalton result is sharp by exhibiting non-locally convex quasi Banach spaces $X$ with a basis for which $C(I,\,X)$ and $C(\Delta ,\, X)$ are isomorphic. Our examples are rather specific and actually, in all cases, $X$ is isomorphic to $C(K,\,X)$ if $K$ is a metric compactum of finite covering dimension.

scholarly journals Proximunalty in Orlicz-bochner Function Spaces

2003 ◽  
Vol 34 (1) ◽  
pp. 71-76 ◽  
Author(s):  
M. Khandaqji ◽  
R. Khalil ◽  
D. Hussein
Keyword(s):  
Banach Space ◽  
Closed Subspace ◽  
Function Spaces ◽  
Unit Interval

A (closed) subspace $ Y$ of a Banach space $ X$ is called proximinal if for every $ x\in X$ there exists some $ y\in Y$ such that $ \|x-y\|\le\|x-z\|$ for $ z\in Y$. It is the object of this paper is to study the proximinality of $ L^\Phi(I,Y)$ in $ L^\Phi(I,X)$ for some class of Young's functions $ \Phi$, where $ I$ is the unit interval. We prove (among other results) that if $ Y$ is a separable proximinal subspace of $ X$, then $ L^\Phi(I,Y)$ is proximinal in $ L^\Phi(I,X)$.

Computing on the Banach space C [ 0 , 1 ]

Computability ◽  
2021 ◽  
pp. 1-14
Author(s):  
Tyler A. Brown
Keyword(s):  
Banach Space ◽  
Unit Interval ◽  
Computable Presentation

We demonstrate that, within any computable presentation of the Banach space C [ 0 , 1 ], computing 1 is no harder than computing the halting set. Additionally, we prove that the modulus operator | · | is Ø ″ -computable and use this to show that C [ 0 , 1 ] is Δ 3 0 -categorical when we restrict ourselves to the presentations in which at least one homeomorphism of the unit interval onto itself is computable.

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