mathlib3 documentation


Units of a normed algebra #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file is a stub, containing a construction of the charted space structure on the group of units of a complete normed ring R, and of the smooth manifold structure on the group of units of a complete normed 𝕜-algebra R.

This manifold is actually a Lie group, which eventually should be the main result of this file.

An important special case of this construction is the general linear group. For a normed space V over a field 𝕜, the 𝕜-linear endomorphisms of V are a normed 𝕜-algebra (see continuous_linear_map.to_normed_algebra), so this construction provides a Lie group structure on its group of units, the general linear group GL(𝕜, V).


The Lie group instance requires the following fields:

instance : lie_group 𝓘(𝕜, R) Rˣ :=
{ smooth_mul := sorry,
  smooth_inv := sorry,
  ..units.smooth_manifold_with_corners }

The ingredients needed for the construction are