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Porosity and the bounded linear regularity property

Reich S., Zaslavski A. J.

Journal of Applied Analysis

2014

Vol. 20, nr 1

1--6

H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.

On the Torricellian point in inner product spaces

Dragomir S., Comanescu D.

Demonstratio Mathematica

2008

Vol. 41, nr 3

639-650

The concept of Torricellian point related to a set of n vectors in normed linear spaces is introduced and the general properties obtained. The existence and uniqueness of the Torricellian point in inner product spaces are established.

A closedness theorem for normed spaces

Poppe H.

Demonstratio Mathematica

2008

Vol. 41, nr 1

123-127

For spaces X, Y, for which some algebraic operations are defined and in some cases topologies for X, Y are defined too, we define for the space X a dual space Xd with respect to the space Y. If [..] is a topology for Y (compatible with the algebraic operations of Y), then the pointwise topology rp for Yx is defined. We show that Xd is (algebraically)rp-closed in Yx. For normed spaces is shown that suitable subspaces of Xd are rp-closed in a product space K C Yx. As a corollary we obtain a generalization of Alaoglu's theorem.

Stability of the Euler-Lagrange-Rassias functional equation

Pietrzyk A.

Demonstratio Mathematica

2006

Vol. 39, nr 3

523-530

Let F be a field, a1, a2 is an element of F, K is an element of {R, C}, s an element of K\{0,1}, X be a linear space over F, S C is contained in X be nonempty, and Y be a Banach space over K. Under some additional assumptions on S we show some stability results for the functional equation Q (a1x + a2y) + Q (a2X - a1y) = s[Q{x) + Q{y)} in the class of function Q : S -> Y.