Search results for "banach"
showing 10 items of 326 documents
The Ptolemy and Zbăganu constants of normed spaces
2010
Abstract In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖ x − y ‖ ‖ z − w ‖ ≤ ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ for any points w , x , y , z in H . It is known that for each normed space ( X , ‖ ⋅ ‖ ) , there exists a constant C such that for any w , x , y , z ∈ X , we have ‖ x − y ‖ ‖ z − w ‖ ≤ C ( ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ ) . The smallest such C is called the Ptolemy constant of X and is denoted by C P ( X ) . We study the relationships between this constant and the geometry of the space X , and hence with metric fix…
Dissipative operators and differential equations on Banach spaces
1991
If we consider the initial value problem Inline Equation $$x'(t) = f(t,x(t)),{\text{ }}x(0) = {x_0}$$ on the real line, it is well known that one—sided bounds like Inline Equation $$\left[ {f(t,x) - f\left( {t,y} \right)} \right]\left( {x - {\text{y}}} \right) \leqslant \omega {\left( {x - y} \right)^2}$$ give much better information about the behaviour of solutions than the Lipschitz- type estimates Inline Equation $$ \left| {f\left( {t,x} \right) - f\left( {t,y} \right)} \right| \leqslant L\left| {x - y} \right|,$$ because ω, unlike L, may be negative.
Some Aspects of Vector-Valued Singular Integrals
2009
Let A, B be Banach spaces and \(1 < p < \infty. \; T\) is said to be a (p, A, B)- CalderoLon–Zygmund type operator if it is of weak type (p, p), and there exist a Banach space E, a bounded bilinear map \(u: E \times A \rightarrow B,\) and a locally integrable function k from \(\mathbb{R}^n \times \mathbb{R}^n \backslash \{(x, x): x \in \mathbb{R}^n\}\) into E such that $$T\;f(x) = \int u(k(x, y), f(y))dy$$ for every A-valued simple function f and \(x \notin \; supp \; f.\)
The Bourgain property and convex hulls
2007
Let (Ω, Σ, μ) be a complete probability space and let X be a Banach space. We consider the following problem: Given a function f: Ω X for which there is a norming set B ⊂ BX * such that Zf,B = {x * ○ f: x * ∈ B } is uniformly integrable and has the Bourgain property, does it follow that f is Birkhoff integrable? It turns out that this question is equivalent to the following one: Given a pointwise bounded family ℋ ⊂ ℝΩ with the Bourgain property, does its convex hull co(ℋ) have the Bourgain property? With the help of an example of D. H. Fremlin, we make clear that both questions have negative answer in general. We prove that a function f: Ω X is scalarly measurable provided that there is a n…
Type and Cotype in Vector-Valued Nakano Sequence Spaces
2001
AbstractGiven a sequence of Banach spaces {Xn}n and a sequence of real numbers {pn}n in [1,∞), the vector-valued Nakano sequence spaces ℓ({pn},{Xn}) consist of elements {xn}n in ∏nXn for which there is a constant λ>0 such that ∑n(‖xn‖/λ)pn<∞. In this paper we find the conditions on the Banach spaces Xn and on the sequence {pn}n for the spaces ℓ({pn},{Xn}) to have cotype q or type p.
Complete weights andv-peak points of spaces of weighted holomorphic functions
2006
We examine the geometric theory of the weighted spaces of holomorphic functions on bounded open subsets ofC n ,C n ,H v (U) and\(H_{v_o } (U)\), by finding a lower bound for the set of weak*-exposed and weak*-strongly exposed points of the unit ball of\(H_{v_o } (U)'\) and give necessary and sufficient conditions for this set to be naturally homeomorphic toU. We apply these results to examine smoothness and strict convexity of\(H_{v_o } (U)\) and\(H_v (U)\). We also investigate whether\(H_{v_o } (U)\) is a dual space.
Fréchet Spaces of Holomorphic Functions without Copies of l 1
1996
Let X be a Banach space. Let Hw*(X*) the Frechet space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, …, of the closed unit ball B(X*) of X* are continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of Hw*(X*) in mB(X*), m = 1,2,…. In this paper we construct the bidual of l1 when this space contains no copy of l1. We also show that if X is an Asplund space, then Hw*(X*) can be represented as the projective limit of a sequence of Banach spaces that are Asplund.
On the Unit Ball of Operator-valued H 2-functions
2009
Let X be a complex Banach space and let H 2 (\( \mathbb{D} \), X) denote the space of X-valued analytic functions in the unit disc such that $$ \mathop {sup}\limits_{0 < r < 1} \int_0^{2\pi } {\left\| {F\left( {re^{it} } \right)} \right\|^2 \frac{{dt}} {{2\pi }} < \infty .} $$ It is shown that a function F belongs to the unit ball of H 2 ( \( \mathbb{D} \), X) if and only if there exist f∈H ∞ (\( \mathbb{D} \), X) and ϕ∈H ∞ (\( \mathbb{D} \)) such that $$ \left\| {f\left( z \right)} \right\|^2 + \left| {\varphi \left( z \right)} \right|^2 \leqslant 1 and F\left( z \right) = \frac{{f\left( z \right)}} {{1 - z\varphi \left( z \right)}} $$ for |z| < 1.
A note on Fréchet and approximate subdifferentials of composite functions
1994
The aim of this note is to present in the reflexive Banach space setting a natural and simple proof of the formula of the approximate subdifferential of a composite function.
Third-order iterative methods without using any Fréchet derivative
2003
AbstractA modification of classical third-order methods is proposed. The main advantage of these methods is they do not need to evaluate any Fréchet derivative. A convergence theorem in Banach spaces, just assuming the second divided difference is bounded and a punctual condition, is analyzed. Finally, some numerical results are presented.