Measure, Integration & Real Analysis · Problem 10A.7

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · inverse-map-continuous

Problem

Suppose

V

is a Hilbert space and

\operatorname{Inv}(V)

is the set of invertible bounded operators on

V

. Think of

\operatorname{Inv}(V)

as a metric space with the metric it inherits as a subset of

\mathcal{B}(V)

. Show that

T \mapsto T^{-1}

is a continuous function from

\operatorname{Inv}(V)

\operatorname{Inv}(V)