On Non-midpoint Locally Uniformly Rotund Normability in Banach Spaces

We will show that if X is a tree-complete subspace of ∞ , which contains c 0 , then it does not admit any weakly midpoint locally uniformly convex renorming. It follows that such a space has no equivalent Kadec renorming. 1. Introduction. It is known that ∞ has an equivalent strictly convex renorming [2]; however, by a result due to Lindenstrauss, it cannot be equivalently renormed in locally uniformly convex manner [10]. In this note, we will show that every tree-complete subspace of ∞ , which contains c 0 , does not admit any equivalent weakly midpoint locally uniformly convex norm. This can be considered as an extension of [1, 8]. Since every strictly convexifiable Banach space with Kadec property admits an equivalent midpoint locally uniformly convex renorming [9], it follows that every subspace of ∞ with the tree-completeness property has no equivalent Kadec renorming. The existence of such a (nontrivial) subspace, which does not contain any copy of ∞ , has already been proved by Haydon and Zizler (see [5, 7]).

Proof. Let ||| · ||| be an equivalent norm on X. We will show that this norm is not wMLUR. Let Choose an element f ( ) of X such that |||f ( ) ||| > (3M ( ) + m ( ) )/4. Then select two disjoint infinite subsets N 0 and N 1 of N \ supp(f ( ) ) with 1 N i ∈ X for some k i ∈ N i , define N i = N i \{k i }, and let Suppose that for some t ∈ T , with |t| < n, A t is specified. Put for each t ∈ T . (b) k ti ∈ N t \ N ti and f t (k t ) = 0 for t ∈ T and i = 0, 1. (c) |||f t ||| > (3M t +m t )/4, where M t and m t denote the supremum and infimum of {|||f ||| : f ∈ A t }, respectively. (d) N s ⊂ N t whenever t ≺ s and N t ∩ N s = ∅, if s and t are not comparable.
is a disjoint family of elements of X. By the tree-completeness of X, there exists some b ∈ {0, 1} N such that ..} and h n = 1 En . By (a) and Next, select some µ ∈ X * , such that µ(h 1 ) = 1 and µ(g) = 0 for each g ∈ c 0 . Clearly, for such an element µ and each n ∈ N, we have µ(h n ) = 1. By and so Therefore, (2.10) The above relations show that This shows that X does not admit any wMLUR norm.
It is known that weakly midpoint locally uniformly rotundity of a Banach space X is equivalent to saying that every point of S(X) is an extreme point of B(X * * ) [11]. It follows that the space considered in Theorem 2.1 has no equivalent norm such that S(X) is a subset of B(X * * ).
A norm on a Banach space X is said to be strictly convex (rotund) (R) if the unit sphere of X contains no nontrivial line segment. We say that a norm is Kadec if the weak and norm topologies coincide on the unit sphere. Every MLUR Banach space admits Kadec renorming (see [1]). Haydon in [6, Corollary 6.6] gives an example of a Kadec renormable space which has no equivalent R norm. The following result gives an example of a strictly convexifiable space with no equivalent Kadec norm. Proof. It is known that ∞ admits an equivalent strictly convex norm (see [4, page 120] or [2]). In [9] it is shown that every R Banach space with the Kadec property admits an equivalent MLUR renorming (see also [3,chapter IV]). Thus the result follows from Theorem 2.1.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation