Real Normed Linear Space Research Articles

Totally bounded sets of precompact linear operators

1. Introduction. In this paper several characterizations of totally bounded sets of precompact operators are given. These lead to an affirmative solution of the conjecture that a collectively compact set A is totally bounded iff Si*= {K*: KCSi} is collectively compact. Let 3E and g) be real or complex normed linear spaces with adjoints i* and D*, respectively. Let the closed unit ball in any of these spaces be denoted by attaching a subscript 1, and let [i, g)] be the set of bounded linear operators with domain i and range in $. An operator A£[2E, g)] is called precompact [compact] iff KHi — {Kx: xCXi} is totally bounded [has a compact closure in the norm topology]. If ?) is complete (i.e., a Banach space), then an operator in [i, §)] is precompact iff it is compact, and a subset Si of [x §)] is totally bounded iff its closure is compact. A subset Si of [X, §)] is called collectively compact iff $& = {Kx : K C $ ; x C ii} has compact closure. There is a detailed theory [3] of strongly convergent sequences in a collectively compact set. Characterizing those collectively compact sets which are totally bounded is thus of interest since a strongly (or weakly) convergent sequence converges in norm iff it is contained in a totally bounded set. Investigation of this problem suggested the statements of Theo

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).

The geometry of Banach spaces. Smoothness

Introduction. This paper contains the first unified treatment of the dual theory of differentiability of the norm functional in a real normed linear space. With this, the work of Smulian [2; 3] is extended and it is shown how uniform convexity is to be modified so as to obtain geometric properties dual to the various types of differentiability of the norm thus answering a question implicit in the work of Lovaglia [1] and Anderson [1]. The resulting dual theory of differentiability is then used to obtain a connection between the differentiability of certain infinite dimensional manifolds imbedded in an infinite dimensional real normed linear space and the continuity properties of the Gaussian spherical image map generalized to such infinite dimensional spaces. In this way a problem proposed by Klee [2, p. 35] is solved. The principal tools employed are (a) Mazur's characterization [1] of a supporting hyperplane of the unit ball as an inverse image of the derivative of the norm functional; (b) a modification of an integral calculus technique given in Krasnosel'skir and RutickiT [1, p. 187]; and (c) James' criterion for the reflexivity of a Banach space, viz., that every continuous linear functional attains its supremum on the unit ball. James [1] has given a proof of this criterion for separable Banach spaces and in James [2] has removed the condition that the Banach space be separable. The paper is divided into five sections. ?1 contains the localization and directionalization of uniform convexity of Clarkson [1] and of full k-convexity of Fan and Glicksberg [1]. ?2, while containing interesting facts in its own right, is motivational in nature for ?3. ?3 contains in the dual theory of differentiability of the norm one of the two main conclusions of the paper. The other main conclusion contained in ?4 is the analysis of the differentiability of norm functionals using the generalized Gaussian spherical image map. The last section, ?5, examines and compares the present results in the context of the geometry of Banach spaces as created and perfected by other workers in the field.