Area orthogonality in normed linear spaces

A new concept of orthogonality in real normed linear spaces is introduced. Typical properties of orthogonality (homogeneity, symmetry, additivity, ...) and relations between this orthogonality and other known orthogonalities (Birkhoff, Boussouis, Unitary-Boussouis and Diminnie) are studied. In particular, some characterizations of inner product spaces are obtained.     

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.     

An orthogonality in normed linear spaces based on angular distance inequality

In this paper, we present a new orthogonality in a normed linear space which is based on an angular distance inequality. Some properties of this orthogonality are discussed. We also find a new approach to the Singer orthogonality in terms of an angular distance inequality. Some related geometric properties of normed linear spaces are discussed. Finally a characterization of inner product spaces is obtained.     

Mappings approximately preserving orthogonality in normed spaces

We answer many open questions regarding approximately orthogonality preserving mappings (in Birkhoff-James sense) in normed spaces. In particular, we show that every approximately orthogonality preserving linear mapping (in Chmieliński sense) is necessarily a scalar multiple of an ε-isometry. Thus, whenever ε-isometries are close to isometries we obtain stability. An example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries, that is, stability does not hold.     

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.     

Preservers of generalized orthogonality on finite dimensional real vector and projective spaces

We describe the general form of bijective orthogonality preserving maps on ? ⁿ equipped with a pair of generalized indefinite inner products. The relations between the projective space and vector space versions of this result are examined and an example is given showing that the hypotheses of our main theorem are essential.     

A characterization of smooth normed linear spaces

In this paper a new characterization of smooth normed linear spaces is discussed using the notion of proximal points of a pair of convex sets. It is proved that a normed linear space is smooth if and only if for each pair of convex sets, points which are mutually nearest to each other from the respective sets are proximal.     

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)     

Nonlinear programming in normed linear spaces

Necessary conditions in the form of multiplier rules are given for a function to have a constrained minimum. First-order differentiability conditions are imposed, and various combinations of set, equality, and inequality constraints are considered in arbitrary normed linear spaces.This paper is based upon part of the author's doctoral dissertation at Ohio University, Athens, Ohio.     

Strong unicity in normed linear spaces

We consider best approximations in a real or complex normed linear space by elements of a finite dimensional subspace. It is the purpose of this paper to characterize, when a best approximation to a given element is strongly unique.     

The differentiability of real functions on normed linear space using generalized subgradients

The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.