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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
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