Definition talk:Invertible Bounded Linear Transformation

I think this presentation overlooks a common subtlety - that "invertible" in the context of a bounded linear transformation means bijective (and so invertible as a regular old function) with bounded inverse. For linear transformations on Banach spaces, inverses of bounded linear transformations are automatically bounded (which is a theorem to be proved in due course), but this is not necessary in general. (it requires $\norm {T x}_Y/\norm x_X$ to be bounded below by some $m > 0$ for $x \ne 0$, whereas generally this might get arbitrarily close to 0) I don't really like this, so will probably say "invertible as a bounded linear transformation" with a link to this page, which I'll rework to make this subtlety clear. (if "bijective with bounded inverse" rings a bell more I can swap it out for that) Caliburn (talk) 11:15, 11 February 2022 (UTC)