On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

On nonconvex retracts in normed linear spaces

Let E be a real normed linear space. A subset X ⊂ E is called a retract of E if there exists a continuous mapping r : E → X, a retraction, satisfying r(x) = x, x ∈ X. It is well known that every nonempty closed convex subset of E is a retract of E. Nonconvex retracts are studied in this paper.

Summable Family in a Commutative Group

Summary Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.

On additive correspondences

The existence of additive selections of additive correspondences was investigated in [Ark. Mat. 4 (1960), 87–97], [Rev. Roumaine Math. Pures Appl. 28 (1983), 239–242.], [Math. Ser. Univ. Novi Sad 18 (1988), 143–148] and other papers. In this article, we find an existence theorem for additive selections of additive correspondences with convex compact values in a real normed linear space defined on an open convex cone of a real separable normed space.

Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Suppose thatEis a real normed linear space,Cis a nonempty convex subset ofE,T:C→Cis a Lipschitzian mapping, andx*∈Cis a fixed point ofT. For givenx0∈C, suppose that the sequence{xn}⊂Cis the Mann iterative sequence defined byxn+1=(1-αn)xn+αnTxn,n≥0, where{αn}is a sequence in [0, 1],∑n=0∞αn2<∞,∑n=0∞αn=∞. We prove that the sequence{xn}strongly converges tox*if and only if there exists a strictly increasing functionΦ:[0,∞)→[0,∞)withΦ(0)=0such thatlimsup n→∞inf j(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.

Characterizations of inner product spaces by means of norm one points

Let $X$ be a a real normed linear space of dimension at least three, with unit sphere $S_X$. In this paper we prove that $X$ is an inner product space if and only if every three point subset of $S_X$ has a Chebyshev center in its convex hull. We also give other characterizations expressed in terms of centers of three point subsets of $S_X$ only. We use in these characterizations Chebyshev centers as well as Fermat centers and $p$-centers.

A distance function property implying differentiability

In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gâteaux (uniformly Fréchet) differentiable norm, then d is Gâteaux (Fréchet) differentiable at x ∈ X/K if there exists an such thatand is Géteaux (Fréchet) differentiable on X / K if there exists a set P+(K) dense in X/K where such a limit is approached uniformly for all x ∈ P+(K). When X is complete this last property implies that K is convex.

Linear Normed Spaces with Extension Property

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).