scholarly journals Local uniform convexity and Kadec-Klee type properties inK-interpolation spaces I: General Theory

2004 ◽  
Vol 2 (2) ◽  
pp. 125-173 ◽  
Author(s):  
Peter G. Dodds ◽  
Theresa K. Dodds ◽  
Alexander A. Sedaev ◽  
Fyodor A. Sukochev
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Systematic Study ◽  
Interpolation Space ◽  
Uniform Convexity ◽  
Interpolation Spaces ◽  
Uniformly Convex ◽  
Property A ◽  
Banach Couple ◽  
Convex Norm

We present a systematic study of the interpolation of local uniform convexity and Kadec-Klee type properties inK-interpolation spaces. Using properties of theK-functional of J.Peetre, our approach is based on a detailed analysis of properties of a Banach couple and properties of aK-interpolation functional which guarantee that a givenK-interpolation space is locally uniformly convex, or has a Kadec-Klee property. A central motivation for our study lies in the observation that classical renorming theorems of Kadec and of Davis, Ghoussoub and Lindenstrauss have an interpolation nature. As a partiular by-product of our study, we show that the theorem of Kadec itself, that each separable Banach space admits an equivalent locally uniformly convex norm, follows directly from our approach.

Asymptotic centres of filters on uniformly rotund spaces

1976 ◽  
Vol 15 (1) ◽  
pp. 87-96
Author(s):  
John Staples
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Convex Banach Space ◽  
Bounded Sequence ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Filter Base

The notion of asymptotic centre of a bounded sequence of points in a uniformly convex Banach space was introduced by Edelstein in order to prove, in a quasi-constructive way, fixed point theorems for nonexpansive and similar maps.Similar theorems have also been proved by, for example, adding a compactness hypothesis to the restrictions on the domain of the maps. In such proofs, which are generally less constructive, it may be possible to weaken the uniform convexity hypothesis.In this paper Edelstein's technique is extended by defining a notion of asymptotic centre for an arbitrary set of nonempty bounded subsets of a metric space. It is shown that when the metric space is uniformly rotund and complete, and when the set of bounded subsets is a filter base, this filter base has a unique asymptotic centre. This fact is used to derive, in a uniform way, several fixed point theorems for nonexpansive and similar maps, both single-valued and many-valued.Though related to known results, each of the fixed point theorems proved is either stronger than the corresponding known result, or has a compactness hypothesis replaced by the assumption of uniform convexity.

scholarly journals Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups

2019 ◽  
pp. 1-47 ◽  
Author(s):  
Masato Mimura ◽  
Hiroki Sako
Keyword(s):  
Banach Space ◽  
Large Scale ◽  
Disjoint Union ◽  
Metric Spaces ◽  
Convex Banach Space ◽  
Linear Groups ◽  
Uniformly Convex ◽  
Property A ◽  
Special Linear Groups ◽  
Accumulation Points

The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In Part I, we prove that a disjoint union has property A of Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems of generators that look similar such that one has property A and the other does not admit (fibered) coarse embeddings into any Banach space with nontrivial type (for instance, any uniformly convex Banach space).

A note on k-uniformly convex spaces

1985 ◽  
Vol 97 (3) ◽  
pp. 489-490
Author(s):  
Jong Sook Bae ◽  
Sung Kyu Choi
Keyword(s):  
Banach Space ◽  
Short Note ◽  
Uniform Convexity ◽  
Uniformly Convex ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

AbstractIn this short note we prove that Istrǎƫescu's notion of k-uniform (k-locally uniform) convexity of a Banach space is actually equivalent to the notion of uniform (locally uniform) convexity. Thus theorem 2 in [3] and theorem 2·6·28 in [2] are trivially true.

scholarly journals Extremal characterizations of reflexive spaces

2007 ◽  
Vol 82 (3) ◽  
pp. 429-440
Author(s):  
Xianfu Wang
Keyword(s):  
Banach Space ◽  
Nonsmooth Analysis ◽  
Necessary Condition ◽  
Uniformly Convex ◽  
Extremal Principle ◽  
Fréchet Differentiable ◽  
Convex Norm

AbstractAssume that a Banach space has a Fréchet differentiable and locally uniformly convex norm. We show that the reflexive property of the Banach space is not only sufficient, but also a necessary condition for the fulfillment of the proximal extremal principle in nonsmooth analysis.

scholarly journals p-Uniform Convexity andq-Uniform Smoothness of Absolute Normalized Norms onℂ2

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Tomonari Suzuki
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Uniform Smoothness

We first prove characterizations ofp-uniform convexity andq-uniform smoothness. We next give a formulation on absolute normalized norms onℂ2. Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is notp-uniformly convex.

Uniform Convexity and the Bishop–Phelps–Bollobás Property

2014 ◽  
Vol 66 (2) ◽  
pp. 373-386 ◽  
Author(s):  
Sun Kwang Kim ◽  
Han Ju Lee
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Uniform Convexity ◽  
Bilinear Forms ◽  
Uniformly Convex

AbstractA new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop–Phelps–Bollobás theorem. It is also proved that the couple of Banach spaces (X;Y) has the Bishop–Phelps–Bollobás property for every Banach space Y when X is uniformly convex. As a corollary, we show that the Bishop–Phelps–Bollobás theorem holds for bilinear forms on ℓp × ℓq (1 < p; q < ∞).

On the centres of hereditary JBW-subalgebras of a JBW-algebra

1979 ◽  
Vol 85 (2) ◽  
pp. 317-324 ◽  
Author(s):  
C. M. Edwards
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Jordan Algebra ◽  
Identity Element ◽  
Image Position

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions thatandfor all elements a and b in A. It follows from (1.1) and (l.2) thatfor all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition thatfor all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

scholarly journals Approximation of positive operators by analytic vectors

2020 ◽  
Vol 12 (2) ◽  
pp. 412-418
Author(s):  
M.I. Dmytryshyn
Keyword(s):  
Banach Space ◽  
Positive Operator ◽  
Invariant Subspaces ◽  
Positive Operators ◽  
Interpolation Spaces ◽  
Jackson Type ◽  
Normalization Factor ◽  
Approximation Spaces ◽  
Similar Role ◽  
Approximation Errors

We give the estimates of approximation errors while approximating of a positive operator $A$ in a Banach space by analytic vectors. Our main results are formulated in the form of Bernstein and Jackson type inequalities with explicitly calculated constants. We consider the classes of invariant subspaces ${\mathcal E}_{q,p}^{\nu,\alpha}(A)$ of analytic vectors of $A$ and the special scale of approximation spaces $\mathcal {B}_{q,p,\tau}^{s,\alpha}(A)$ associated with the complex degrees of positive operator. The approximation spaces are determined by $E$-functional, that plays a similar role as the module of smoothness. We show that the approximation spaces can be considered as interpolation spaces generated by $K$-method of real interpolation. The constants in the Bernstein and Jackson type inequalities are expressed using the normalization factor.

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