Boundedness and Invertibility

Abstract

Boundedness is a useful and natural condition, but it is a very strong condition on a linear transformation. The condition has a profound effect throughout operator theory, from its mildest algebraic aspects to its most complicated topological ones. To avoid certain obvious mistakes, it is important to know that boundedness is more than just the conjunction of an infinite number of conditions, one for each element of a basis. If A is an operator on a Hilbert space H with an orthonormal basis {e₁, e₂, e₃,•••}, thenthenumbers ‖Ae _n‖ arebounded; if, forinstance, ‖A‖ ≦ 1, then ‖Ae _n‖ ≦ 1 for all n; and, of course, if A = 0, then Ae _n = 0 for all n. The obvious mistakes just mentioned are based on the assumption that the converses of these assertions are true.