# Units of a normed algebra #

We construct the Lie group structure on the group of units of a complete normed š-algebra R. The group of units RĖ£ has a natural smooth manifold structure modelled on R given by its embedding into R. Together with the smoothness of the multiplication and inverse of its elements, RĖ£ forms a Lie group.

An important special case of this construction is the general linear group. For a normed space V over a field š, the š-linear endomorphisms of V are a normed š-algebra (see ContinuousLinearMap.toNormedAlgebra), so this construction provides a Lie group structure on its group of units, the general linear group GL(š, V), as demonstrated by:

example {V : Type*} [NormedAddCommGroup V] [NormedSpace š V] [CompleteSpace V] :
LieGroup š(š, V āL[š] V) (V āL[š] V)Ė£ := inferInstance

instance Units.instChartedSpace {R : Type u_1} [] [] :
Equations
• Units.instChartedSpace = āÆ.singletonChartedSpace
theorem Units.chartAt_apply {R : Type u_1} [] [] {a : RĖ£} {b : RĖ£} :
ā(chartAt R a) b = āb
theorem Units.chartAt_source {R : Type u_1} [] [] {a : RĖ£} :
(chartAt R a).source = Set.univ
instance Units.instSmoothManifoldWithCornersModelWithCornersSelf {R : Type u_1} [] [] {š : Type u_2} [] [NormedAlgebra š R] :
Equations
• āÆ = āÆ
theorem Units.contMDiff_val {R : Type u_1} [] [] {š : Type u_2} [] [NormedAlgebra š R] {m : āā} :
ContMDiff (modelWithCornersSelf š R) (modelWithCornersSelf š R) m Units.val

For a complete normed ring R, the embedding of the units RĖ£ into R is a smooth map between manifolds.

instance Units.instLieGroupModelWithCornersSelf {R : Type u_1} [] [] {š : Type u_2} [] [NormedAlgebra š R] :

The units of a complete normed ring form a Lie group.

Equations
• āÆ = āÆ