Characterization of Compact Operators in Pre-Hilbert and Hilbert Spaces

Abstract

The concept of a compact operator on a Hilbert space, H is an extension of the concept of a matrix acting on a finite-dimensional vector space. In Hilbert space, compact operators are precisely the closure of finite rank operators in the topology induced by the operator norm. In this paper, we provide an elementary exposition of compact linear operators in pre-Hilbert and Hilbert spaces. However, whenever advantageous, we may prove a few results in the general context of normed linear spaces. It is well known that strong convergence implies weak convergence but weak convergence does not imply strong convergence. We also show that an operator T \(\epsilon\) B(H) is compact if and only if T maps every weakly convergent sequence in H to a strongly convergent sequence.

On Some Aspects of Compactness in Metric Spaces

“…A metric space (X, ρ) is said to be totally bounded (or precompact) if, for every ϵ > 0, the space X can be covered by a finite family of open balls of radius ϵ [5]. A metric space X is said to be sequentially compact if every sequence (x n ) ∞ n=1 of points in X has a convergent subsequence [6]. This abstracts the Bolzano-Weierstrass property, that is, closed bounded subsets of the real line are sequentially compact.…”