Units of a normed algebra #

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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).

TODO #

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

• smoothness of multiplication and inversion in the charts, i.e. as functions on the normed 𝕜-space R: see cont_diff_at_ring_inverse for the inversion result, and cont_diff_mul (needs to be generalized from field to algebra) for the multiplication result
• for an open embedding f, whose domain is equipped with the induced manifold structure f.singleton_smooth_manifold_with_corners, characterization of smoothness of functions to/from this manifold in terms of smoothness in the target space. See the pair of lemmas cont_mdiff_coe_sphere and cont_mdiff.cod_restrict_sphere for a model. None of this should be particularly difficult.