Tietze Extension Theorem for n-dimensional Spaces

Summary In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of En with a non-empty interior. This article is based on [20]; [23] and [22] can also serve as reference books.

Brouwer Fixed Point Theorem in the General Case

Brouwer Fixed Point Theorem in the General Case In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of εn with a non empty interior. This article is based on [15].

Smoothness in spaces of compact operators

We prove that if X and Y are two (real) Banach spaces such that dim X ≥ 2 and dim Y ≥ 2, then the space K(X, Y) contains a convex compact subset C with dim C ≥ 2 (in the affine sense) which fails to be an intersection of balls. This improves two results of Ruess and Stegall.