Best coapproximation in normed linear spaces

We consider, in normed linear spaces, a kind of approximation by elements of linear subspaces, introduced byC. Franchetti andM. Furi [5], which we call best coapproximation. We obtain some results on characterization and existence of elements of best coapproximation in arbitrary normed linear spaces and in spaces of continuous functions. We give some characterizations of strict convexity in terms of best coapproximation and we study some properties of the setvalued operators of best coapproximation.Work performed partially under the auspices of the GNAFA (National Group for Functional Analysis and its Applications) of the CNR (National Research Council of Italy)     

Best coapproximation in quotient spaces

As a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi. In this paper, we apply the above coapproximation, and obtain some results on the upper semi-continuity of cometric projection maps. Also we shall determine under what conditions coproximinality can be transmitted to and from quotient spaces.     

The metric gradient in normed linear spaces

Summary In this paper we give a definition for the gradient of a functional defined on a normed linear space which in the case of Hilbert space reduces to the usual definition. We also establish some interesting properties of the gradient which allow us to extend the well-known theorem of Curry to a large class of normed linear spaces.A part of this research was sponsored by the U.S. Army under contract No. DA-31-124-ARO-D-462.     

Uniform linear approximation in normed linear spaces

It has been shown that the theory of H-sets is important in the characterization of best uniform approximation of continuous real-or complex-valued functions. We here extend the theory of H-sets to the more general setting of functions with compact domain and with range contained in a Banach space. Using the definitions of H-sets, we construct a maximal linear functional and obtain inclusion theorems analogous to the classical case. It is then a simple matter to deduce a characterization of best approximation and show when uniqueness and strong uniqueness are achieved.     

Minimal linear spaces

A decent linear space DLS(k) is a linear space with minimal line size at least three and with maximal line size exactly k. Denote by v_k (resp. b_k) the minimum number of points (resp. lines) in a DLS(k). We determine the numbers v_k and b_k for all k and prove that each DLS(k) with b_k lines has v_k points. Thus the DLS(k)'s with b_k lines are the minimal linear spaces.     

The non-existence of certain finite linear spaces

We study the existence of finite linear spaces with v points and n ²+n+2 lines, where n ²+1?v?n ²+n+1. For n?3, there is only one such linear space; it has ten points and fourteen lines.     

Connectedness in topological linear spaces

The following properties, well known for normed linear spaces of dimension ≧2, are established for an arbitrary topological linear space of dimension ≧2: (a) every neighborhood of 0 contains one whose complement is connected; (b) the complement of a bounded set has exactly one unbounded component. Research supported by the National Science Foundation, U.S.A. (NSF-GP-378).     

Measuring flatness in linear spaces

Let L be a finite-dimensional normed linear space and let M be a compact subset of L lying on one side of a hyperplane through 0. A measure of flatness for M is the number $\text{D(M) =}\text{inf}\text{{}\text{sup}\text{f(x)}\text{f(y)}\text{: x, y ? M}}$, where the infimum is over all f in $\text{L}{}^1$ which are positive on M. Thus D(M) = 1 if M is flat, but otherwise D(M) > 1. On the other hand, let E(M) be a second measure on M defined as follows: If M is linearly independent, E(M) = 1. If M is linearly dependent, then (1) let Z be a minimal, linearly dependent subset of M; (2) partition Z into mutually exclusive subsets U = {u₁, …, u_p} and V = {v₁, …, v_q} such that there exist positive coefficients a_i and b_i for which Σ_{i = 1}^pa_iu_i = Σ_{i = 1}^qb_iv_i; (3) let $\text{r =}\text{max}\text{{}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{a}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{b}{}_{\mathrm{i}}\text{,}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{b}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{a}{}_{\mathrm{i}}\text{}}$; (4) let E(M) be the supremum of all ratios r which can be formed by steps (1), (2) and (3). The main result of this paper is that these two measures are the same: D(M) = E(M). This result is then used to obtain results concerning the Banach distance-coefficient between an arbitrary finite-dimensional normed linear space and Hilbert space.     

The Aleksandrov problem in linear 2-normed spaces

We introduce the concept of 2-isometry which is suitable to represent the notion of area preserving mappings in linear 2-normed spaces. And then we obtain some results for the Aleksandrov problem in linear 2-normed spaces.     

The Goldstine Theorem for asymmetric normed linear spaces

It is well known that if (X,q) is an asymmetric normed linear space, then the function qs defined on X by qs(x)=max{q(x),q(−x)}, is a norm on the linear space X. However, the lack of symmetry in the definition of the asymmetric norm q yields an algebraic asymmetry in the dual space of (X,q). This fact establishes a significant difference with the standard results on duality that hold in the case of locally convex spaces. In this paper we study some aspects of a reflexivity theory in the setting of asymmetric normed linear spaces. In particular, we obtain a version of the Goldstine Theorem to these spaces which is applied to prove, among other results, a characterization of reflexive asymmetric normed linear spaces.     

The contraction linear metric projections inL p spaces

In this paper, we study the contraction linearity for metric projection in Lp spaces. A geometrical property of a subspace Y of Lp is given on which Pr is a contraction projection.