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/- | |
Copyright (c) 2021 Anatole Dedecker. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Anatole Dedecker, Eric Wieser | |
-/ | |
import analysis.specific_limits.basic | |
import analysis.analytic.basic | |
import analysis.complex.basic | |
import data.nat.choose.cast | |
import data.finset.noncomm_prod | |
/-! | |
# Exponential in a Banach algebra | |
In this file, we define `exp π : πΈ β πΈ`, the exponential map in a topological algebra `πΈ` over a | |
field `π`. | |
While for most interesting results we need `πΈ` to be normed algebra, we do not require this in the | |
definition in order to make `exp` independent of a particular choice of norm. The definition also | |
does not require that `πΈ` be complete, but we need to assume it for most results. | |
We then prove some basic results, but we avoid importing derivatives here to minimize dependencies. | |
Results involving derivatives and comparisons with `real.exp` and `complex.exp` can be found in | |
`analysis/special_functions/exponential`. | |
## Main results | |
We prove most result for an arbitrary field `π`, and then specialize to `π = β` or `π = β`. | |
### General case | |
- `exp_add_of_commute_of_mem_ball` : if `π` has characteristic zero, then given two commuting | |
elements `x` and `y` in the disk of convergence, we have | |
`exp π (x+y) = (exp π x) * (exp π y)` | |
- `exp_add_of_mem_ball` : if `π` has characteristic zero and `πΈ` is commutative, then given two | |
elements `x` and `y` in the disk of convergence, we have | |
`exp π (x+y) = (exp π x) * (exp π y)` | |
- `exp_neg_of_mem_ball` : if `π` has characteristic zero and `πΈ` is a division ring, then given an | |
element `x` in the disk of convergence, we have `exp π (-x) = (exp π x)β»ΒΉ`. | |
### `π = β` or `π = β` | |
- `exp_series_radius_eq_top` : the `formal_multilinear_series` defining `exp π` has infinite | |
radius of convergence | |
- `exp_add_of_commute` : given two commuting elements `x` and `y`, we have | |
`exp π (x+y) = (exp π x) * (exp π y)` | |
- `exp_add` : if `πΈ` is commutative, then we have `exp π (x+y) = (exp π x) * (exp π y)` | |
for any `x` and `y` | |
- `exp_neg` : if `πΈ` is a division ring, then we have `exp π (-x) = (exp π x)β»ΒΉ`. | |
- `exp_sum_of_commute` : the analogous result to `exp_add_of_commute` for `finset.sum`. | |
- `exp_sum` : the analogous result to `exp_add` for `finset.sum`. | |
- `exp_nsmul` : repeated addition in the domain corresponds to repeated multiplication in the | |
codomain. | |
- `exp_zsmul` : repeated addition in the domain corresponds to repeated multiplication in the | |
codomain. | |
### Other useful compatibility results | |
- `exp_eq_exp` : if `πΈ` is a normed algebra over two fields `π` and `π'`, then `exp π = exp π' πΈ` | |
-/ | |
open filter is_R_or_C continuous_multilinear_map normed_field asymptotics | |
open_locale nat topological_space big_operators ennreal | |
section topological_algebra | |
variables (π πΈ : Type*) [field π] [ring πΈ] [algebra π πΈ] [topological_space πΈ] | |
[topological_ring πΈ] | |
/-- `exp_series π πΈ` is the `formal_multilinear_series` whose `n`-th term is the map | |
`(xα΅’) : πΈβΏ β¦ (1/n! : π) β’ β xα΅’`. Its sum is the exponential map `exp π : πΈ β πΈ`. -/ | |
def exp_series : formal_multilinear_series π πΈ πΈ := | |
Ξ» n, (n!β»ΒΉ : π) β’ continuous_multilinear_map.mk_pi_algebra_fin π n πΈ | |
variables {πΈ} | |
/-- `exp π : πΈ β πΈ` is the exponential map determined by the action of `π` on `πΈ`. | |
It is defined as the sum of the `formal_multilinear_series` `exp_series π πΈ`. | |
Note that when `πΈ = matrix n n π`, this is the **Matrix Exponential**; see | |
[`analysis.normed_space.matrix_exponential`](../matrix_exponential) for lemmas specific to that | |
case. -/ | |
noncomputable def exp (x : πΈ) : πΈ := (exp_series π πΈ).sum x | |
variables {π} | |
lemma exp_series_apply_eq (x : πΈ) (n : β) : exp_series π πΈ n (Ξ» _, x) = (n!β»ΒΉ : π) β’ x^n := | |
by simp [exp_series] | |
lemma exp_series_apply_eq' (x : πΈ) : | |
(Ξ» n, exp_series π πΈ n (Ξ» _, x)) = (Ξ» n, (n!β»ΒΉ : π) β’ x^n) := | |
funext (exp_series_apply_eq x) | |
lemma exp_series_sum_eq (x : πΈ) : (exp_series π πΈ).sum x = β' (n : β), (n!β»ΒΉ : π) β’ x^n := | |
tsum_congr (Ξ» n, exp_series_apply_eq x n) | |
lemma exp_eq_tsum : exp π = (Ξ» x : πΈ, β' (n : β), (n!β»ΒΉ : π) β’ x^n) := | |
funext exp_series_sum_eq | |
@[simp] lemma exp_zero [t2_space πΈ] : exp π (0 : πΈ) = 1 := | |
begin | |
suffices : (Ξ» x : πΈ, β' (n : β), (n!β»ΒΉ : π) β’ x^n) 0 = β' (n : β), if n = 0 then 1 else 0, | |
{ have key : β n β ({0} : finset β), (if n = 0 then (1 : πΈ) else 0) = 0, | |
from Ξ» n hn, if_neg (finset.not_mem_singleton.mp hn), | |
rw [exp_eq_tsum, this, tsum_eq_sum key, finset.sum_singleton], | |
simp }, | |
refine tsum_congr (Ξ» n, _), | |
split_ifs with h h; | |
simp [h] | |
end | |
@[simp] lemma exp_op [t2_space πΈ] (x : πΈ) : | |
exp π (mul_opposite.op x) = mul_opposite.op (exp π x) := | |
by simp_rw [exp, exp_series_sum_eq, βmul_opposite.op_pow, βmul_opposite.op_smul, tsum_op] | |
@[simp] lemma exp_unop [t2_space πΈ] (x : πΈα΅α΅α΅) : | |
exp π (mul_opposite.unop x) = mul_opposite.unop (exp π x) := | |
by simp_rw [exp, exp_series_sum_eq, βmul_opposite.unop_pow, βmul_opposite.unop_smul, tsum_unop] | |
lemma star_exp [t2_space πΈ] [star_ring πΈ] [has_continuous_star πΈ] (x : πΈ) : | |
star (exp π x) = exp π (star x) := | |
by simp_rw [exp_eq_tsum, βstar_pow, βstar_inv_nat_cast_smul, βtsum_star] | |
variables (π) | |
lemma commute.exp_right [t2_space πΈ] {x y : πΈ} (h : commute x y) : commute x (exp π y) := | |
begin | |
rw exp_eq_tsum, | |
exact commute.tsum_right x (Ξ» n, (h.pow_right n).smul_right _), | |
end | |
lemma commute.exp_left [t2_space πΈ] {x y : πΈ} (h : commute x y) : commute (exp π x) y := | |
(h.symm.exp_right π).symm | |
lemma commute.exp [t2_space πΈ] {x y : πΈ} (h : commute x y) : commute (exp π x) (exp π y) := | |
(h.exp_left _).exp_right _ | |
end topological_algebra | |
section topological_division_algebra | |
variables {π πΈ : Type*} [field π] [division_ring πΈ] [algebra π πΈ] [topological_space πΈ] | |
[topological_ring πΈ] | |
lemma exp_series_apply_eq_div (x : πΈ) (n : β) : exp_series π πΈ n (Ξ» _, x) = x^n / n! := | |
by rw [div_eq_mul_inv, β(nat.cast_commute n! (x ^ n)).inv_leftβ.eq, βsmul_eq_mul, | |
exp_series_apply_eq, inv_nat_cast_smul_eq _ _ _ _] | |
lemma exp_series_apply_eq_div' (x : πΈ) : (Ξ» n, exp_series π πΈ n (Ξ» _, x)) = (Ξ» n, x^n / n!) := | |
funext (exp_series_apply_eq_div x) | |
lemma exp_series_sum_eq_div (x : πΈ) : (exp_series π πΈ).sum x = β' (n : β), x^n / n! := | |
tsum_congr (exp_series_apply_eq_div x) | |
lemma exp_eq_tsum_div : exp π = (Ξ» x : πΈ, β' (n : β), x^n / n!) := | |
funext exp_series_sum_eq_div | |
end topological_division_algebra | |
section normed | |
section any_field_any_algebra | |
variables {π πΈ πΉ : Type*} [nontrivially_normed_field π] | |
variables [normed_ring πΈ] [normed_ring πΉ] [normed_algebra π πΈ] [normed_algebra π πΉ] | |
lemma norm_exp_series_summable_of_mem_ball (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
summable (Ξ» n, β₯exp_series π πΈ n (Ξ» _, x)β₯) := | |
(exp_series π πΈ).summable_norm_apply hx | |
lemma norm_exp_series_summable_of_mem_ball' (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
summable (Ξ» n, β₯(n!β»ΒΉ : π) β’ x^nβ₯) := | |
begin | |
change summable (norm β _), | |
rw β exp_series_apply_eq', | |
exact norm_exp_series_summable_of_mem_ball x hx | |
end | |
section complete_algebra | |
variables [complete_space πΈ] | |
lemma exp_series_summable_of_mem_ball (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
summable (Ξ» n, exp_series π πΈ n (Ξ» _, x)) := | |
summable_of_summable_norm (norm_exp_series_summable_of_mem_ball x hx) | |
lemma exp_series_summable_of_mem_ball' (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
summable (Ξ» n, (n!β»ΒΉ : π) β’ x^n) := | |
summable_of_summable_norm (norm_exp_series_summable_of_mem_ball' x hx) | |
lemma exp_series_has_sum_exp_of_mem_ball (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
has_sum (Ξ» n, exp_series π πΈ n (Ξ» _, x)) (exp π x) := | |
formal_multilinear_series.has_sum (exp_series π πΈ) hx | |
lemma exp_series_has_sum_exp_of_mem_ball' (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
has_sum (Ξ» n, (n!β»ΒΉ : π) β’ x^n) (exp π x):= | |
begin | |
rw β exp_series_apply_eq', | |
exact exp_series_has_sum_exp_of_mem_ball x hx | |
end | |
lemma has_fpower_series_on_ball_exp_of_radius_pos (h : 0 < (exp_series π πΈ).radius) : | |
has_fpower_series_on_ball (exp π) (exp_series π πΈ) 0 (exp_series π πΈ).radius := | |
(exp_series π πΈ).has_fpower_series_on_ball h | |
lemma has_fpower_series_at_exp_zero_of_radius_pos (h : 0 < (exp_series π πΈ).radius) : | |
has_fpower_series_at (exp π) (exp_series π πΈ) 0 := | |
(has_fpower_series_on_ball_exp_of_radius_pos h).has_fpower_series_at | |
lemma continuous_on_exp : | |
continuous_on (exp π : πΈ β πΈ) (emetric.ball 0 (exp_series π πΈ).radius) := | |
formal_multilinear_series.continuous_on | |
lemma analytic_at_exp_of_mem_ball (x : πΈ) (hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
analytic_at π (exp π) x:= | |
begin | |
by_cases h : (exp_series π πΈ).radius = 0, | |
{ rw h at hx, exact (ennreal.not_lt_zero hx).elim }, | |
{ have h := pos_iff_ne_zero.mpr h, | |
exact (has_fpower_series_on_ball_exp_of_radius_pos h).analytic_at_of_mem hx } | |
end | |
/-- In a Banach-algebra `πΈ` over a normed field `π` of characteristic zero, if `x` and `y` are | |
in the disk of convergence and commute, then `exp π (x + y) = (exp π x) * (exp π y)`. -/ | |
lemma exp_add_of_commute_of_mem_ball [char_zero π] | |
{x y : πΈ} (hxy : commute x y) (hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) | |
(hy : y β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
exp π (x + y) = (exp π x) * (exp π y) := | |
begin | |
rw [exp_eq_tsum, tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm | |
(norm_exp_series_summable_of_mem_ball' x hx) (norm_exp_series_summable_of_mem_ball' y hy)], | |
dsimp only, | |
conv_lhs {congr, funext, rw [hxy.add_pow' _, finset.smul_sum]}, | |
refine tsum_congr (Ξ» n, finset.sum_congr rfl $ Ξ» kl hkl, _), | |
rw [nsmul_eq_smul_cast π, smul_smul, smul_mul_smul, β (finset.nat.mem_antidiagonal.mp hkl), | |
nat.cast_add_choose, (finset.nat.mem_antidiagonal.mp hkl)], | |
congr' 1, | |
have : (n! : π) β 0 := nat.cast_ne_zero.mpr n.factorial_ne_zero, | |
field_simp [this] | |
end | |
/-- `exp π x` has explicit two-sided inverse `exp π (-x)`. -/ | |
noncomputable def invertible_exp_of_mem_ball [char_zero π] {x : πΈ} | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : invertible (exp π x) := | |
{ inv_of := exp π (-x), | |
inv_of_mul_self := begin | |
have hnx : -x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius, | |
{ rw [emetric.mem_ball, βneg_zero, edist_neg_neg], | |
exact hx }, | |
rw [βexp_add_of_commute_of_mem_ball (commute.neg_left $ commute.refl x) hnx hx, neg_add_self, | |
exp_zero], | |
end, | |
mul_inv_of_self := begin | |
have hnx : -x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius, | |
{ rw [emetric.mem_ball, βneg_zero, edist_neg_neg], | |
exact hx }, | |
rw [βexp_add_of_commute_of_mem_ball (commute.neg_right $ commute.refl x) hx hnx, add_neg_self, | |
exp_zero], | |
end } | |
lemma is_unit_exp_of_mem_ball [char_zero π] {x : πΈ} | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : is_unit (exp π x) := | |
@is_unit_of_invertible _ _ _ (invertible_exp_of_mem_ball hx) | |
lemma inv_of_exp_of_mem_ball [char_zero π] {x : πΈ} | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) [invertible (exp π x)] : | |
β (exp π x) = exp π (-x) := | |
by { letI := invertible_exp_of_mem_ball hx, convert (rfl : β (exp π x) = _) } | |
/-- Any continuous ring homomorphism commutes with `exp`. -/ | |
lemma map_exp_of_mem_ball {F} [ring_hom_class F πΈ πΉ] (f : F) (hf : continuous f) (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
f (exp π x) = exp π (f x) := | |
begin | |
rw [exp_eq_tsum, exp_eq_tsum], | |
refine ((exp_series_summable_of_mem_ball' _ hx).has_sum.map f hf).tsum_eq.symm.trans _, | |
dsimp only [function.comp], | |
simp_rw [one_div, map_inv_nat_cast_smul f π π, map_pow], | |
end | |
end complete_algebra | |
lemma algebra_map_exp_comm_of_mem_ball [complete_space π] (x : π) | |
(hx : x β emetric.ball (0 : π) (exp_series π π).radius) : | |
algebra_map π πΈ (exp π x) = exp π (algebra_map π πΈ x) := | |
map_exp_of_mem_ball _ (algebra_map_clm _ _).continuous _ hx | |
end any_field_any_algebra | |
section any_field_division_algebra | |
variables {π πΈ : Type*} [nontrivially_normed_field π] [normed_division_ring πΈ] [normed_algebra π πΈ] | |
variables (π) | |
lemma norm_exp_series_div_summable_of_mem_ball (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
summable (Ξ» n, β₯x^n / n!β₯) := | |
begin | |
change summable (norm β _), | |
rw β exp_series_apply_eq_div' x, | |
exact norm_exp_series_summable_of_mem_ball x hx | |
end | |
lemma exp_series_div_summable_of_mem_ball [complete_space πΈ] (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : summable (Ξ» n, x^n / n!) := | |
summable_of_summable_norm (norm_exp_series_div_summable_of_mem_ball π x hx) | |
lemma exp_series_div_has_sum_exp_of_mem_ball [complete_space πΈ] (x : πΈ) | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : has_sum (Ξ» n, x^n / n!) (exp π x) := | |
begin | |
rw β exp_series_apply_eq_div' x, | |
exact exp_series_has_sum_exp_of_mem_ball x hx | |
end | |
variables {π} | |
lemma exp_neg_of_mem_ball [char_zero π] [complete_space πΈ] {x : πΈ} | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
exp π (-x) = (exp π x)β»ΒΉ := | |
begin | |
letI := invertible_exp_of_mem_ball hx, | |
exact inv_of_eq_inv (exp π x), | |
end | |
end any_field_division_algebra | |
section any_field_comm_algebra | |
variables {π πΈ : Type*} [nontrivially_normed_field π] [normed_comm_ring πΈ] [normed_algebra π πΈ] | |
[complete_space πΈ] | |
/-- In a commutative Banach-algebra `πΈ` over a normed field `π` of characteristic zero, | |
`exp π (x+y) = (exp π x) * (exp π y)` for all `x`, `y` in the disk of convergence. -/ | |
lemma exp_add_of_mem_ball [char_zero π] {x y : πΈ} | |
(hx : x β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) | |
(hy : y β emetric.ball (0 : πΈ) (exp_series π πΈ).radius) : | |
exp π (x + y) = (exp π x) * (exp π y) := | |
exp_add_of_commute_of_mem_ball (commute.all x y) hx hy | |
end any_field_comm_algebra | |
section is_R_or_C | |
section any_algebra | |
variables (π πΈ πΉ : Type*) [is_R_or_C π] [normed_ring πΈ] [normed_algebra π πΈ] | |
variables [normed_ring πΉ] [normed_algebra π πΉ] | |
/-- In a normed algebra `πΈ` over `π = β` or `π = β`, the series defining the exponential map | |
has an infinite radius of convergence. -/ | |
lemma exp_series_radius_eq_top : (exp_series π πΈ).radius = β := | |
begin | |
refine (exp_series π πΈ).radius_eq_top_of_summable_norm (Ξ» r, _), | |
refine summable_of_norm_bounded_eventually _ (real.summable_pow_div_factorial r) _, | |
filter_upwards [eventually_cofinite_ne 0] with n hn, | |
rw [norm_mul, norm_norm (exp_series π πΈ n), exp_series, norm_smul, norm_inv, norm_pow, | |
nnreal.norm_eq, norm_eq_abs, abs_cast_nat, mul_comm, βmul_assoc, βdiv_eq_mul_inv], | |
have : β₯continuous_multilinear_map.mk_pi_algebra_fin π n πΈβ₯ β€ 1 := | |
norm_mk_pi_algebra_fin_le_of_pos (nat.pos_of_ne_zero hn), | |
exact mul_le_of_le_one_right (div_nonneg (pow_nonneg r.coe_nonneg n) n!.cast_nonneg) this | |
end | |
lemma exp_series_radius_pos : 0 < (exp_series π πΈ).radius := | |
begin | |
rw exp_series_radius_eq_top, | |
exact with_top.zero_lt_top | |
end | |
variables {π πΈ πΉ} | |
lemma norm_exp_series_summable (x : πΈ) : summable (Ξ» n, β₯exp_series π πΈ n (Ξ» _, x)β₯) := | |
norm_exp_series_summable_of_mem_ball x ((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
lemma norm_exp_series_summable' (x : πΈ) : summable (Ξ» n, β₯(n!β»ΒΉ : π) β’ x^nβ₯) := | |
norm_exp_series_summable_of_mem_ball' x ((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
section complete_algebra | |
variables [complete_space πΈ] | |
lemma exp_series_summable (x : πΈ) : summable (Ξ» n, exp_series π πΈ n (Ξ» _, x)) := | |
summable_of_summable_norm (norm_exp_series_summable x) | |
lemma exp_series_summable' (x : πΈ) : summable (Ξ» n, (n!β»ΒΉ : π) β’ x^n) := | |
summable_of_summable_norm (norm_exp_series_summable' x) | |
lemma exp_series_has_sum_exp (x : πΈ) : has_sum (Ξ» n, exp_series π πΈ n (Ξ» _, x)) (exp π x) := | |
exp_series_has_sum_exp_of_mem_ball x ((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
lemma exp_series_has_sum_exp' (x : πΈ) : has_sum (Ξ» n, (n!β»ΒΉ : π) β’ x^n) (exp π x):= | |
exp_series_has_sum_exp_of_mem_ball' x ((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
lemma exp_has_fpower_series_on_ball : | |
has_fpower_series_on_ball (exp π) (exp_series π πΈ) 0 β := | |
exp_series_radius_eq_top π πΈ βΈ | |
has_fpower_series_on_ball_exp_of_radius_pos (exp_series_radius_pos _ _) | |
lemma exp_has_fpower_series_at_zero : | |
has_fpower_series_at (exp π) (exp_series π πΈ) 0 := | |
exp_has_fpower_series_on_ball.has_fpower_series_at | |
lemma exp_continuous : continuous (exp π : πΈ β πΈ) := | |
begin | |
rw [continuous_iff_continuous_on_univ, β metric.eball_top_eq_univ (0 : πΈ), | |
β exp_series_radius_eq_top π πΈ], | |
exact continuous_on_exp | |
end | |
lemma exp_analytic (x : πΈ) : | |
analytic_at π (exp π) x := | |
analytic_at_exp_of_mem_ball x ((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
/-- In a Banach-algebra `πΈ` over `π = β` or `π = β`, if `x` and `y` commute, then | |
`exp π (x+y) = (exp π x) * (exp π y)`. -/ | |
lemma exp_add_of_commute | |
{x y : πΈ} (hxy : commute x y) : | |
exp π (x + y) = (exp π x) * (exp π y) := | |
exp_add_of_commute_of_mem_ball hxy ((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
section | |
variables (π) | |
/-- `exp π x` has explicit two-sided inverse `exp π (-x)`. -/ | |
noncomputable def invertible_exp (x : πΈ) : invertible (exp π x) := | |
invertible_exp_of_mem_ball $ (exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _ | |
lemma is_unit_exp (x : πΈ) : is_unit (exp π x) := | |
is_unit_exp_of_mem_ball $ (exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _ | |
lemma inv_of_exp (x : πΈ) [invertible (exp π x)] : | |
β (exp π x) = exp π (-x) := | |
inv_of_exp_of_mem_ball $ (exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _ | |
lemma ring.inverse_exp (x : πΈ) : ring.inverse (exp π x) = exp π (-x) := | |
begin | |
letI := invertible_exp π x, | |
exact ring.inverse_invertible _, | |
end | |
end | |
/-- In a Banach-algebra `πΈ` over `π = β` or `π = β`, if a family of elements `f i` mutually | |
commute then `exp π (β i, f i) = β i, exp π (f i)`. -/ | |
lemma exp_sum_of_commute {ΞΉ} (s : finset ΞΉ) (f : ΞΉ β πΈ) | |
(h : β (i β s) (j β s), commute (f i) (f j)) : | |
exp π (β i in s, f i) = s.noncomm_prod (Ξ» i, exp π (f i)) | |
(Ξ» i hi j hj, (h i hi j hj).exp π) := | |
begin | |
classical, | |
induction s using finset.induction_on with a s ha ih, | |
{ simp }, | |
rw [finset.noncomm_prod_insert_of_not_mem _ _ _ _ ha, finset.sum_insert ha, | |
exp_add_of_commute, ih], | |
refine commute.sum_right _ _ _ _, | |
intros i hi, | |
exact h _ (finset.mem_insert_self _ _) _ (finset.mem_insert_of_mem hi), | |
end | |
lemma exp_nsmul (n : β) (x : πΈ) : | |
exp π (n β’ x) = exp π x ^ n := | |
begin | |
induction n with n ih, | |
{ rw [zero_smul, pow_zero, exp_zero], }, | |
{ rw [succ_nsmul, pow_succ, exp_add_of_commute ((commute.refl x).smul_right n), ih] } | |
end | |
variables (π) | |
/-- Any continuous ring homomorphism commutes with `exp`. -/ | |
lemma map_exp {F} [ring_hom_class F πΈ πΉ] (f : F) (hf : continuous f) (x : πΈ) : | |
f (exp π x) = exp π (f x) := | |
map_exp_of_mem_ball f hf x $ (exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _ | |
lemma exp_smul {G} [monoid G] [mul_semiring_action G πΈ] [has_continuous_const_smul G πΈ] | |
(g : G) (x : πΈ) : | |
exp π (g β’ x) = g β’ exp π x := | |
(map_exp π (mul_semiring_action.to_ring_hom G πΈ g) (continuous_const_smul _) x).symm | |
lemma exp_units_conj (y : πΈΛ£) (x : πΈ) : | |
exp π (y * x * β(yβ»ΒΉ) : πΈ) = y * exp π x * β(yβ»ΒΉ) := | |
exp_smul _ (conj_act.to_conj_act y) x | |
lemma exp_units_conj' (y : πΈΛ£) (x : πΈ) : | |
exp π (β(yβ»ΒΉ) * x * y) = β(yβ»ΒΉ) * exp π x * y := | |
exp_units_conj _ _ _ | |
@[simp] lemma prod.fst_exp [complete_space πΉ] (x : πΈ Γ πΉ) : (exp π x).fst = exp π x.fst := | |
map_exp _ (ring_hom.fst πΈ πΉ) continuous_fst x | |
@[simp] lemma prod.snd_exp [complete_space πΉ] (x : πΈ Γ πΉ) : (exp π x).snd = exp π x.snd := | |
map_exp _ (ring_hom.snd πΈ πΉ) continuous_snd x | |
@[simp] lemma pi.exp_apply {ΞΉ : Type*} {πΈ : ΞΉ β Type*} [fintype ΞΉ] | |
[Ξ i, normed_ring (πΈ i)] [Ξ i, normed_algebra π (πΈ i)] [Ξ i, complete_space (πΈ i)] | |
(x : Ξ i, πΈ i) (i : ΞΉ) : | |
exp π x i = exp π (x i) := | |
begin | |
-- Lean struggles to infer this instance due to it wanting `[Ξ i, semi_normed_ring (πΈ i)]` | |
letI : normed_algebra π (Ξ i, πΈ i) := pi.normed_algebra _, | |
exact map_exp _ (pi.eval_ring_hom πΈ i) (continuous_apply _) x | |
end | |
lemma pi.exp_def {ΞΉ : Type*} {πΈ : ΞΉ β Type*} [fintype ΞΉ] | |
[Ξ i, normed_ring (πΈ i)] [Ξ i, normed_algebra π (πΈ i)] [Ξ i, complete_space (πΈ i)] | |
(x : Ξ i, πΈ i) : | |
exp π x = Ξ» i, exp π (x i) := | |
funext $ pi.exp_apply π x | |
lemma function.update_exp {ΞΉ : Type*} {πΈ : ΞΉ β Type*} [fintype ΞΉ] [decidable_eq ΞΉ] | |
[Ξ i, normed_ring (πΈ i)] [Ξ i, normed_algebra π (πΈ i)] [Ξ i, complete_space (πΈ i)] | |
(x : Ξ i, πΈ i) (j : ΞΉ) (xj : πΈ j) : | |
function.update (exp π x) j (exp π xj) = exp π (function.update x j xj) := | |
begin | |
ext i, | |
simp_rw [pi.exp_def], | |
exact (function.apply_update (Ξ» i, exp π) x j xj i).symm, | |
end | |
end complete_algebra | |
lemma algebra_map_exp_comm (x : π) : | |
algebra_map π πΈ (exp π x) = exp π (algebra_map π πΈ x) := | |
algebra_map_exp_comm_of_mem_ball x $ | |
(exp_series_radius_eq_top π π).symm βΈ edist_lt_top _ _ | |
end any_algebra | |
section division_algebra | |
variables {π πΈ : Type*} [is_R_or_C π] [normed_division_ring πΈ] [normed_algebra π πΈ] | |
variables (π) | |
lemma norm_exp_series_div_summable (x : πΈ) : summable (Ξ» n, β₯x^n / n!β₯) := | |
norm_exp_series_div_summable_of_mem_ball π x | |
((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
variables [complete_space πΈ] | |
lemma exp_series_div_summable (x : πΈ) : summable (Ξ» n, x^n / n!) := | |
summable_of_summable_norm (norm_exp_series_div_summable π x) | |
lemma exp_series_div_has_sum_exp (x : πΈ) : has_sum (Ξ» n, x^n / n!) (exp π x):= | |
exp_series_div_has_sum_exp_of_mem_ball π x | |
((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
variables {π} | |
lemma exp_neg (x : πΈ) : exp π (-x) = (exp π x)β»ΒΉ := | |
exp_neg_of_mem_ball $ (exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _ | |
lemma exp_zsmul (z : β€) (x : πΈ) : exp π (z β’ x) = (exp π x) ^ z := | |
begin | |
obtain β¨n, rfl | rflβ© := z.eq_coe_or_neg, | |
{ rw [zpow_coe_nat, coe_nat_zsmul, exp_nsmul] }, | |
{ rw [zpow_neg, zpow_coe_nat, neg_smul, exp_neg, coe_nat_zsmul, exp_nsmul] }, | |
end | |
lemma exp_conj (y : πΈ) (x : πΈ) (hy : y β 0) : | |
exp π (y * x * yβ»ΒΉ) = y * exp π x * yβ»ΒΉ := | |
exp_units_conj _ (units.mk0 y hy) x | |
lemma exp_conj' (y : πΈ) (x : πΈ) (hy : y β 0) : | |
exp π (yβ»ΒΉ * x * y) = yβ»ΒΉ * exp π x * y := | |
exp_units_conj' _ (units.mk0 y hy) x | |
end division_algebra | |
section comm_algebra | |
variables {π πΈ : Type*} [is_R_or_C π] [normed_comm_ring πΈ] [normed_algebra π πΈ] [complete_space πΈ] | |
/-- In a commutative Banach-algebra `πΈ` over `π = β` or `π = β`, | |
`exp π (x+y) = (exp π x) * (exp π y)`. -/ | |
lemma exp_add {x y : πΈ} : exp π (x + y) = (exp π x) * (exp π y) := | |
exp_add_of_mem_ball ((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
((exp_series_radius_eq_top π πΈ).symm βΈ edist_lt_top _ _) | |
/-- A version of `exp_sum_of_commute` for a commutative Banach-algebra. -/ | |
lemma exp_sum {ΞΉ} (s : finset ΞΉ) (f : ΞΉ β πΈ) : | |
exp π (β i in s, f i) = β i in s, exp π (f i) := | |
begin | |
rw [exp_sum_of_commute, finset.noncomm_prod_eq_prod], | |
exact Ξ» i hi j hj, commute.all _ _, | |
end | |
end comm_algebra | |
end is_R_or_C | |
end normed | |
section scalar_tower | |
variables (π π' πΈ : Type*) [field π] [field π'] [ring πΈ] [algebra π πΈ] [algebra π' πΈ] | |
[topological_space πΈ] [topological_ring πΈ] | |
/-- If a normed ring `πΈ` is a normed algebra over two fields, then they define the same | |
`exp_series` on `πΈ`. -/ | |
lemma exp_series_eq_exp_series (n : β) (x : πΈ) : | |
(exp_series π πΈ n (Ξ» _, x)) = (exp_series π' πΈ n (Ξ» _, x)) := | |
by rw [exp_series_apply_eq, exp_series_apply_eq, inv_nat_cast_smul_eq π π'] | |
/-- If a normed ring `πΈ` is a normed algebra over two fields, then they define the same | |
exponential function on `πΈ`. -/ | |
lemma exp_eq_exp : (exp π : πΈ β πΈ) = exp π' := | |
begin | |
ext, | |
rw [exp, exp], | |
refine tsum_congr (Ξ» n, _), | |
rw exp_series_eq_exp_series π π' πΈ n x | |
end | |
lemma exp_β_β_eq_exp_β_β : (exp β : β β β) = exp β := | |
exp_eq_exp β β β | |
end scalar_tower | |