If [itex]g(a) \neq 0[/itex] and both [itex]f[/itex] and [itex]g[/itex] are continuous at [itex]a[/itex], then we know the quotient function [itex]f/g[/itex] is continuous at [itex]a[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Now, suppose we have a linear operator [itex]A(t)[/itex] on a Hilbert space such that the function [itex]\phi(t) = \| A(t) \|[/itex], [itex]\phi: \mathbb R \to [0,\infty)[/itex], is continuous at [itex]a[/itex]. Do we then know that the function [itex]\varphi(t) = \|A(t)^{-1}\|[/itex], [itex]\varphi: \mathbb R \to [0,\infty)[/itex] is continuous at [itex]a[/itex], provided the inverse exists there? Any ideas on how to tackle this question?

I guess I should add that [itex]A(t)[/itex] is a family of bounded linear operators depending on a continuous real parameter [itex]t[/itex].

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Continuity of the inverse of a linear operator

Loading...

Similar Threads - Continuity inverse linear | Date |
---|---|

I Differentialbility & Continuity of Multivariable Functions | Feb 20, 2018 |

I Existence of Partial Derivatives and Continuity ... | Feb 17, 2018 |

I Continuity of the determinant function | Sep 26, 2017 |

I Noncompact locally compact Hausdorff continuous mapping | Sep 23, 2017 |

A continuous function having an inverse <=> conditions on a derivative? | Jan 16, 2013 |

**Physics Forums - The Fusion of Science and Community**