Collectively Compact Sets of Gradient Mappings

Gradient Mappings

Constructive Compact Linear Mappings

In this paper, we deal with compact linear mappings of a normed linear space, within the framework of Bishop's constructive mathematics. We prove the constructive substitutes for the classically well-known theorems on compact linear mappings: T is compact if and only if T* is compact; if S is bounded and if T is compact, then TS is compact; if S and T is compact, then S+ T is compact.

Compact Sets and Compact Operators

Proof. Properties 2 and 3 are left to the reader. For property 1, assume that S is an unbounded compact set. Since S is unbounded, we may select a sequence {vn}n=1 such that ‖vn‖ → 0 as n→∞. Since S is compact, this sequence will have a convergent subsequence, say {vk}k=1, which will still be unbounded. This sequence is Cauchy, so there is a positive integer K for which ‖v`− vm‖ ≤ 1/2 for all `...

Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces

Sets, Infinity, and Mappings

These notes discuss sets, mappings between sets, and the notion of countably infinite. This material supplements the lecture on countably and uncountably infinite sets in the EE 503 probability class. The material is useful because probability theory is defined over abstract sets, probabilities are defined as measures on subsets, and random variables are defined by mappings from an abstract set...