# mathlibdocumentation

analysis.normed_space.exponential

# Exponential in a Banach algebra #

In this file, we define exp 𝕂 𝔸, the exponential map in a normed algebra 𝔸 over a nondiscrete normed field 𝕂. Although the definition doesn't require 𝔸 to be complete, we need to assume it for most results.

We then prove basic results, as described below.

## Main result #

We prove most result for an arbitrary field 𝕂, and then specialize to 𝕂 = ℝ or 𝕂 = ℂ.

### General case #

• has_strict_fderiv_at_exp_zero_of_radius_pos : exp 𝕂 𝔸 has strict Fréchet-derivative 1 : 𝔸 →L[𝕂] 𝔸 at zero, as long as it converges on a neighborhood of zero (see also has_strict_deriv_at_exp_zero_of_radius_pos for the case 𝔸 = 𝕂)
• exp_add_of_commute_of_lt_radius : if 𝕂 has characteristic zero, then given two commuting elements x and y in the disk of convergence, we have exp 𝕂 𝔸 (x+y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y)
• exp_add_of_lt_radius : if 𝕂 has characteristic zero and 𝔸 is commutative, then given two elements x and y in the disk of convergence, we have exp 𝕂 𝔸 (x+y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y)
• has_strict_fderiv_at_exp_of_lt_radius : if 𝕂 has characteristic zero and 𝔸 is commutative, then given a point x in the disk of convergence, exp 𝕂 𝔸 as strict Fréchet-derivative exp 𝕂 𝔸 x • 1 : 𝔸 →L[𝕂] 𝔸 at x (see also has_strict_deriv_at_exp_of_lt_radius for the case 𝔸 = 𝕂)

### 𝕂 = ℝ or 𝕂 = ℂ#

• exp_series_radius_eq_top : the formal_multilinear_series defining exp 𝕂 𝔸 has infinite radius of convergence
• has_strict_fderiv_at_exp_zero : exp 𝕂 𝔸 has strict Fréchet-derivative 1 : 𝔸 →L[𝕂] 𝔸 at zero (see also has_strict_deriv_at_exp_zero for the case 𝔸 = 𝕂)
• exp_add_of_commute : given two commuting elements x and y, we have exp 𝕂 𝔸 (x+y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y)
• exp_add : if 𝔸 is commutative, then we have exp 𝕂 𝔸 (x+y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y) for any x and y
• has_strict_fderiv_at_exp : if 𝔸 is commutative, then given any point x, exp 𝕂 𝔸 as strict Fréchet-derivative exp 𝕂 𝔸 x • 1 : 𝔸 →L[𝕂] 𝔸 at x (see also has_strict_deriv_at_exp for the case 𝔸 = 𝕂)

### Other useful compatibility results #

• exp_eq_exp : if 𝔸 is a normed algebra over two fields 𝕂 and 𝕂', then exp 𝕂 𝔸 = exp 𝕂' 𝔸
• complex.exp_eq_exp_ℂ_ℂ : complex.exp = exp ℂ ℂ
• real.exp_eq_exp_ℝ_ℝ : real.exp = exp ℝ ℝ
def exp_series (𝕂 : Type u_1) (𝔸 : Type u_2) [normed_ring 𝔸] [ 𝔸] :

In a Banach algebra 𝔸 over a normed field 𝕂, exp_series 𝕂 𝔸 is the formal_multilinear_series whose n-th term is the map (xᵢ) : 𝔸ⁿ ↦ (1/n! : 𝕂) • ∏ xᵢ. Its sum is the exponential map exp 𝕂 𝔸 : 𝔸 → 𝔸.

Equations
def exp (𝕂 : Type u_1) (𝔸 : Type u_2) [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
𝔸

In a Banach algebra 𝔸 over a normed field 𝕂, exp 𝕂 𝔸 : 𝔸 → 𝔸 is the exponential map determined by the action of 𝕂 on 𝔸. It is defined as the sum of the formal_multilinear_series exp_series 𝕂 𝔸.

Equations
theorem exp_series_apply_eq {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (n : ) :
𝔸 n) (λ (_x : fin n), x) = (1 / n!) x ^ n
theorem exp_series_apply_eq' {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
(λ (n : ), 𝔸 n) (λ (_x : fin n), x)) = λ (n : ), (1 / n!) x ^ n
theorem exp_series_apply_eq_field {𝕂 : Type u_1} (x : 𝕂) (n : ) :
𝕂 n) (λ (_x : fin n), x) = x ^ n / n!
theorem exp_series_apply_eq_field' {𝕂 : Type u_1} (x : 𝕂) :
(λ (n : ), 𝕂 n) (λ (_x : fin n), x)) = λ (n : ), x ^ n / n!
theorem exp_series_sum_eq {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
𝔸).sum x = ∑' (n : ), (1 / n!) x ^ n
theorem exp_series_sum_eq_field {𝕂 : Type u_1} (x : 𝕂) :
𝕂).sum x = ∑' (n : ), x ^ n / n!
theorem exp_eq_tsum {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] :
exp 𝕂 𝔸 = λ (x : 𝔸), ∑' (n : ), (1 / n!) x ^ n
theorem exp_eq_tsum_field {𝕂 : Type u_1}  :
exp 𝕂 𝕂 = λ (x : 𝕂), ∑' (n : ), x ^ n / n!
theorem exp_zero {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] :
exp 𝕂 𝔸 0 = 1
theorem norm_exp_series_summable_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (hx : x 𝔸).radius) :
summable (λ (n : ), 𝔸 n) (λ (_x : fin n), x))
theorem norm_exp_series_summable_of_mem_ball' {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (hx : x 𝔸).radius) :
summable (λ (n : ), (1 / n!) x ^ n)
theorem norm_exp_series_field_summable_of_mem_ball {𝕂 : Type u_1} (x : 𝕂) (hx : x 𝕂).radius) :
summable (λ (n : ), x ^ n / n!)
theorem exp_series_summable_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (hx : x 𝔸).radius) :
summable (λ (n : ), 𝔸 n) (λ (_x : fin n), x))
theorem exp_series_summable_of_mem_ball' {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (hx : x 𝔸).radius) :
summable (λ (n : ), (1 / n!) x ^ n)
theorem exp_series_field_summable_of_mem_ball {𝕂 : Type u_1} (x : 𝕂) (hx : x 𝕂).radius) :
summable (λ (n : ), x ^ n / n!)
theorem exp_series_has_sum_exp_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (hx : x 𝔸).radius) :
has_sum (λ (n : ), 𝔸 n) (λ (_x : fin n), x)) (exp 𝕂 𝔸 x)
theorem exp_series_has_sum_exp_of_mem_ball' {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (hx : x 𝔸).radius) :
has_sum (λ (n : ), (1 / n!) x ^ n) (exp 𝕂 𝔸 x)
theorem exp_series_field_has_sum_exp_of_mem_ball {𝕂 : Type u_1} (x : 𝕂) (hx : x 𝕂).radius) :
has_sum (λ (n : ), x ^ n / n!) (exp 𝕂 𝕂 x)
theorem has_fpower_series_on_ball_exp_of_radius_pos {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (h : 0 < 𝔸).radius) :
theorem has_fpower_series_at_exp_zero_of_radius_pos {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (h : 0 < 𝔸).radius) :
has_fpower_series_at (exp 𝕂 𝔸) 𝔸) 0
theorem continuous_on_exp {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸]  :
theorem analytic_at_exp_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (x : 𝔸) (hx : x 𝔸).radius) :
(exp 𝕂 𝔸) x
theorem has_strict_fderiv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (h : 0 < 𝔸).radius) :
has_strict_fderiv_at (exp 𝕂 𝔸) 1 0

The exponential in a Banach-algebra 𝔸 over a normed field 𝕂 has strict Fréchet-derivative 1 : 𝔸 →L[𝕂] 𝔸 at zero, as long as it converges on a neighborhood of zero.

theorem has_fderiv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] (h : 0 < 𝔸).radius) :
has_fderiv_at (exp 𝕂 𝔸) 1 0

The exponential in a Banach-algebra 𝔸 over a normed field 𝕂 has Fréchet-derivative 1 : 𝔸 →L[𝕂] 𝔸 at zero, as long as it converges on a neighborhood of zero.

theorem exp_add_of_commute_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [normed_ring 𝔸] [ 𝔸] [char_zero 𝕂] {x y : 𝔸} (hxy : y) (hx : x 𝔸).radius) (hy : y 𝔸).radius) :
exp 𝕂 𝔸 (x + y) = (exp 𝕂 𝔸 x) * exp 𝕂 𝔸 y

In a Banach-algebra 𝔸 over a normed field 𝕂 of characteristic zero, if x and y are in the disk of convergence and commute, then exp 𝕂 𝔸 (x + y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y).

theorem exp_add_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [ 𝔸] [char_zero 𝕂] {x y : 𝔸} (hx : x 𝔸).radius) (hy : y 𝔸).radius) :
exp 𝕂 𝔸 (x + y) = (exp 𝕂 𝔸 x) * exp 𝕂 𝔸 y

In a commutative Banach-algebra 𝔸 over a normed field 𝕂 of characteristic zero, exp 𝕂 𝔸 (x+y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y) for all x, y in the disk of convergence.

theorem has_fderiv_at_exp_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [ 𝔸] [char_zero 𝕂] {x : 𝔸} (hx : x 𝔸).radius) :
has_fderiv_at (exp 𝕂 𝔸) (exp 𝕂 𝔸 x 1) x

The exponential map in a commutative Banach-algebra 𝔸 over a normed field 𝕂 of characteristic zero has Fréchet-derivative exp 𝕂 𝔸 x • 1 : 𝔸 →L[𝕂] 𝔸 at any point x in the disk of convergence.

theorem has_strict_fderiv_at_exp_of_mem_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [ 𝔸] [char_zero 𝕂] {x : 𝔸} (hx : x 𝔸).radius) :
has_strict_fderiv_at (exp 𝕂 𝔸) (exp 𝕂 𝔸 x 1) x

The exponential map in a commutative Banach-algebra 𝔸 over a normed field 𝕂 of characteristic zero has strict Fréchet-derivative exp 𝕂 𝔸 x • 1 : 𝔸 →L[𝕂] 𝔸 at any point x in the disk of convergence.

theorem has_strict_deriv_at_exp_of_mem_ball {𝕂 : Type u_1} [char_zero 𝕂] {x : 𝕂} (hx : x 𝕂).radius) :
has_strict_deriv_at (exp 𝕂 𝕂) (exp 𝕂 𝕂 x) x

The exponential map in a complete normed field 𝕂 of characteristic zero has strict derivative exp 𝕂 𝕂 x at any point x in the disk of convergence.

theorem has_deriv_at_exp_of_mem_ball {𝕂 : Type u_1} [char_zero 𝕂] {x : 𝕂} (hx : x 𝕂).radius) :
has_deriv_at (exp 𝕂 𝕂) (exp 𝕂 𝕂 x) x

The exponential map in a complete normed field 𝕂 of characteristic zero has derivative exp 𝕂 𝕂 x at any point x in the disk of convergence.

theorem has_strict_deriv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} (h : 0 < 𝕂).radius) :
has_strict_deriv_at (exp 𝕂 𝕂) 1 0

The exponential map in a complete normed field 𝕂 of characteristic zero has strict derivative 1 at zero, as long as it converges on a neighborhood of zero.

theorem has_deriv_at_exp_zero_of_radius_pos {𝕂 : Type u_1} (h : 0 < 𝕂).radius) :
has_deriv_at (exp 𝕂 𝕂) 1 0

The exponential map in a complete normed field 𝕂 of characteristic zero has derivative 1 at zero, as long as it converges on a neighborhood of zero.

theorem exp_series_radius_eq_top (𝕂 : Type u_1) (𝔸 : Type u_2) [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] :

In a normed algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ, the series defining the exponential map has an infinite radius of convergence.

theorem exp_series_radius_pos (𝕂 : Type u_1) (𝔸 : Type u_2) [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] :
theorem norm_exp_series_summable {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
summable (λ (n : ), 𝔸 n) (λ (_x : fin n), x))
theorem norm_exp_series_summable' {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
summable (λ (n : ), (1 / n!) x ^ n)
theorem norm_exp_series_field_summable {𝕂 : Type u_1} [is_R_or_C 𝕂] (x : 𝕂) :
summable (λ (n : ), x ^ n / n!)
theorem exp_series_summable {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
summable (λ (n : ), 𝔸 n) (λ (_x : fin n), x))
theorem exp_series_summable' {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
summable (λ (n : ), (1 / n!) x ^ n)
theorem exp_series_field_summable {𝕂 : Type u_1} [is_R_or_C 𝕂] (x : 𝕂) :
summable (λ (n : ), x ^ n / n!)
theorem exp_series_has_sum_exp {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
has_sum (λ (n : ), 𝔸 n) (λ (_x : fin n), x)) (exp 𝕂 𝔸 x)
theorem exp_series_has_sum_exp' {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
has_sum (λ (n : ), (1 / n!) x ^ n) (exp 𝕂 𝔸 x)
theorem exp_series_field_has_sum_exp {𝕂 : Type u_1} [is_R_or_C 𝕂] (x : 𝕂) :
has_sum (λ (n : ), x ^ n / n!) (exp 𝕂 𝕂 x)
theorem exp_has_fpower_series_on_ball {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸]  :
𝔸) 0
theorem exp_has_fpower_series_at_zero {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸]  :
has_fpower_series_at (exp 𝕂 𝔸) 𝔸) 0
theorem exp_continuous {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸]  :
continuous (exp 𝕂 𝔸)
theorem exp_analytic {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] (x : 𝔸) :
(exp 𝕂 𝔸) x
theorem has_strict_fderiv_at_exp_zero {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸]  :
has_strict_fderiv_at (exp 𝕂 𝔸) 1 0

The exponential in a Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ has strict Fréchet-derivative 1 : 𝔸 →L[𝕂] 𝔸 at zero.

theorem has_fderiv_at_exp_zero {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸]  :
has_fderiv_at (exp 𝕂 𝔸) 1 0

The exponential in a Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ has Fréchet-derivative 1 : 𝔸 →L[𝕂] 𝔸 at zero.

theorem exp_add_of_commute {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [normed_ring 𝔸] [ 𝔸] {x y : 𝔸} (hxy : y) :
exp 𝕂 𝔸 (x + y) = (exp 𝕂 𝔸 x) * exp 𝕂 𝔸 y

In a Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ, if x and y commute, then exp 𝕂 𝔸 (x+y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y).

theorem exp_add {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [ 𝔸] {x y : 𝔸} :
exp 𝕂 𝔸 (x + y) = (exp 𝕂 𝔸 x) * exp 𝕂 𝔸 y

In a comutative Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ, exp 𝕂 𝔸 (x+y) = (exp 𝕂 𝔸 x) * (exp 𝕂 𝔸 y).

theorem has_strict_fderiv_at_exp {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [ 𝔸] {x : 𝔸} :
has_strict_fderiv_at (exp 𝕂 𝔸) (exp 𝕂 𝔸 x 1) x

The exponential map in a commutative Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ has strict Fréchet-derivative exp 𝕂 𝔸 x • 1 : 𝔸 →L[𝕂] 𝔸 at any point x.

theorem has_fderiv_at_exp {𝕂 : Type u_1} {𝔸 : Type u_2} [is_R_or_C 𝕂] [ 𝔸] {x : 𝔸} :
has_fderiv_at (exp 𝕂 𝔸) (exp 𝕂 𝔸 x 1) x

The exponential map in a commutative Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ has Fréchet-derivative exp 𝕂 𝔸 x • 1 : 𝔸 →L[𝕂] 𝔸 at any point x.

theorem has_strict_deriv_at_exp {𝕂 : Type u_1} [is_R_or_C 𝕂] {x : 𝕂} :
has_strict_deriv_at (exp 𝕂 𝕂) (exp 𝕂 𝕂 x) x

The exponential map in 𝕂 = ℝ or 𝕂 = ℂ has strict derivative exp 𝕂 𝕂 x at any point x.

theorem has_deriv_at_exp {𝕂 : Type u_1} [is_R_or_C 𝕂] {x : 𝕂} :
has_deriv_at (exp 𝕂 𝕂) (exp 𝕂 𝕂 x) x

The exponential map in 𝕂 = ℝ or 𝕂 = ℂ has derivative exp 𝕂 𝕂 x at any point x.

theorem has_strict_deriv_at_exp_zero {𝕂 : Type u_1} [is_R_or_C 𝕂] :
has_strict_deriv_at (exp 𝕂 𝕂) 1 0

The exponential map in 𝕂 = ℝ or 𝕂 = ℂ has strict derivative 1 at zero.

theorem has_deriv_at_exp_zero {𝕂 : Type u_1} [is_R_or_C 𝕂] :
has_deriv_at (exp 𝕂 𝕂) 1 0

The exponential map in 𝕂 = ℝ or 𝕂 = ℂ has derivative 1 at zero.

theorem exp_series_eq_exp_series (𝕂 : Type u_1) (𝕂' : Type u_2) (𝔸 : Type u_3) [normed_ring 𝔸] [ 𝔸] [ 𝔸] (n : ) (x : 𝔸) :
𝔸 n) (λ (_x : fin n), x) = (exp_series 𝕂' 𝔸 n) (λ (_x : fin n), x)

If a normed ring 𝔸 is a normed algebra over two fields, then they define the same exp_series on 𝔸.

theorem exp_eq_exp (𝕂 : Type u_1) (𝕂' : Type u_2) (𝔸 : Type u_3) [normed_ring 𝔸] [ 𝔸] [ 𝔸] :
exp 𝕂 𝔸 = exp 𝕂' 𝔸

If a normed ring 𝔸 is a normed algebra over two fields, then they define the same exponential function on 𝔸.