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Topological Properties of Real Normed Space

100%

Nakasho K. , Futa Y. , Shidama Y.

nr 3

209-223

In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on [19](p.3-41), [2] and [34](p.3-67).

Arrangements of series preserving their convergence or boundedness

75%

Drewnowski L.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2007

tom Vol. 47, [Z] 1

57-75

For a map p of N into itself, consider the induced transformation [...] of series in a topological vector space. Then such properties of this transformation as sending convergent series to convergent series, or convergent series to bounded series, or bounded series to bounded series (and a few more) are mutually equivalent. Moreover, they are equivalent to an intrinsic property of p which reduces to those found by Agnew and Pleasants (in the case of permutations) and Witula (in the general case) as necessary and sufficient conditions for the above transformation to preserve convergence of scalar series. In the paper, the scalar case is treated first using simple Banach space methods, and then the result is easily extended to the general setting.

Continuity of superquadratic set-valued functions

75%

Troczka-Pawelec K.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2012

tom Vol. 17

77–86

Let X = (X,+) be an arbitrary topological group. The aim of the paper is to prove a regularity theorem for set-valued superquadratic functions, that is solutions of the inclusion 2F(s) + 2F(t) ⊂ F(s + t) + F(s − t), s, t ∈ X, with values in a topological vector space.