:: Exponential Function on Complex {B}anach Algebra
:: by Noboru Endou
::
:: Received April 6, 2004
:: Copyright (c) 2004-2012 Association of Mizar Users


begin

Lm1:  for X being Complex_Banach_Algebra
for z being Element of X
for n being Element of NAT holds
( z * (z #N n) = z #N (n + 1) & (z #N n) * z = z #N (n + 1) & z * (z #N n) = (z #N n) * z )
proofend;

Lm2:  for X being Complex_Banach_Algebra
for n being Element of NAT
for z, w being Element of X st z,w are_commutative holds
( w * (z #N n) = (z #N n) * w & z * (w #N n) = (w #N n) * z )
proofend;

theorem Th1: :: CLOPBAN4:1
for X being Complex_Banach_Algebra
for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent & lim (seq1 - seq2) = 0. X holds
lim seq1 = lim seq2
proofend;

theorem Th2: :: CLOPBAN4:2
for X being Complex_Banach_Algebra
for s being sequence of X
for z being Element of X st ( for n being Element of NAT holds s . n = z ) holds
lim s = z
proofend;

theorem Th3: :: CLOPBAN4:3
for X being Complex_Banach_Algebra
for s, s9 being sequence of X st s is convergent & s9 is convergent holds
s * s9 is convergent
proofend;

theorem Th4: :: CLOPBAN4:4
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
z * s is convergent
proofend;

theorem Th5: :: CLOPBAN4:5
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
s * z is convergent
proofend;

theorem Th6: :: CLOPBAN4:6
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
lim (z * s) = z * (lim s)
proofend;

theorem Th7: :: CLOPBAN4:7
for X being Complex_Banach_Algebra
for z being Element of X
for s being sequence of X st s is convergent holds
lim (s * z) = (lim s) * z
proofend;

theorem Th8: :: CLOPBAN4:8
for X being Complex_Banach_Algebra
for s, s9 being sequence of X st s is convergent & s9 is convergent holds
lim (s * s9) = (lim s) * (lim s9)
proofend;

theorem Th9: :: CLOPBAN4:9
for X being Complex_Banach_Algebra
for z being Element of X
for seq being sequence of X holds
( Partial_Sums (z * seq) = z * (Partial_Sums seq) & Partial_Sums (seq * z) = (Partial_Sums seq) * z )
proofend;

theorem Th10: :: CLOPBAN4:10
for X being Complex_Banach_Algebra
for k being Element of NAT
for seq being sequence of X holds ||.((Partial_Sums seq) . k).|| <= (Partial_Sums ||.seq.||) . k
proofend;

theorem Th11: :: CLOPBAN4:11
for X being Complex_Banach_Algebra
for m being Element of NAT
for seq1, seq2 being sequence of X st ( for n being Element of NAT st n <= m holds
seq1 . n = seq2 . n ) holds
(Partial_Sums seq1) . m = (Partial_Sums seq2) . m
proofend;

theorem Th12: :: CLOPBAN4:12
for X being Complex_Banach_Algebra
for seq being sequence of X
for rseq being Real_Sequence st ( for n being Element of NAT holds ||.(seq . n).|| <= rseq . n ) &amp; rseq is convergent &amp; lim rseq = 0 holds
( seq is convergent &amp; lim seq = 0. X )
proofend;

definition
let X be Complex_Banach_Algebra;
let z be Element of X;
funcz ExpSeq  ->  sequence of X means :Def1: :: CLOPBAN4:def 1
 for n being Element of NAT holds it . n = (1r / (n !)) * (z #N n);
existence
 ex b1 being sequence of X st
for n being Element of NAT holds b1 . n = (1r / (n !)) * (z #N n)
proofend;
uniqueness
 for b1, b2 being sequence of X st ( for n being Element of NAT holds b1 . n = (1r / (n !)) * (z #N n) ) &amp; ( for n being Element of NAT holds b2 . n = (1r / (n !)) * (z #N n) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def1 defines ExpSeq CLOPBAN4:def_1_:_
 for X being Complex_Banach_Algebra
for z being Element of X
for b3 being sequence of X holds
( b3 = z ExpSeq iff for n being Element of NAT holds b3 . n = (1r / (n !)) * (z #N n) );

scheme :: CLOPBAN4:sch 1
ExNormSpaceCASE{ F1() ->  Complex_Banach_Algebra, F2( Element of NAT , Element of NAT ) ->  Point of F1() } :
 for k being Element of NAT ex seq being sequence of F1() st
for n being Element of NAT holds
( ( n <= k implies seq . n = F2(k,n) ) &amp; ( n > k implies seq . n = 0. F1() ) )
proofend;

definition
let n be Element of NAT ;
let X be Complex_Banach_Algebra;
let z, w be Element of X;
func Expan (n,z,w) ->  sequence of X means :Def2: :: CLOPBAN4:def 2
 for k being Element of NAT holds
( ( k <= n implies it . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies it . k = 0. X ) );
existence
 ex b1 being sequence of X st
for k being Element of NAT holds
( ( k <= n implies b1 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b1 . k = 0. X ) )
proofend;
uniqueness
 for b1, b2 being sequence of X st ( for k being Element of NAT holds
( ( k <= n implies b1 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b1 . k = 0. X ) ) ) &amp; ( for k being Element of NAT holds
( ( k <= n implies b2 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def2 defines Expan CLOPBAN4:def_2_:_
 for n being Element of NAT
for X being Complex_Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Expan (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b5 . k = 0. X ) ) );

definition
let n be Element of NAT ;
let X be Complex_Banach_Algebra;
let z, w be Element of X;
func Expan_e (n,z,w) ->  sequence of X means :Def3: :: CLOPBAN4:def 3
 for k being Element of NAT holds
( ( k <= n implies it . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies it . k = 0. X ) );
existence
 ex b1 being sequence of X st
for k being Element of NAT holds
( ( k <= n implies b1 . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b1 . k = 0. X ) )
proofend;
uniqueness
 for b1, b2 being sequence of X st ( for k being Element of NAT holds
( ( k <= n implies b1 . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b1 . k = 0. X ) ) ) &amp; ( for k being Element of NAT holds
( ( k <= n implies b2 . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def3 defines Expan_e CLOPBAN4:def_3_:_
 for n being Element of NAT
for X being Complex_Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Expan_e (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) &amp; ( n < k implies b5 . k = 0. X ) ) );

definition
let n be Element of NAT ;
let X be Complex_Banach_Algebra;
let z, w be Element of X;
func Alfa (n,z,w) ->  sequence of X means :Def4: :: CLOPBAN4:def 4
 for k being Element of NAT holds
( ( k <= n implies it . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) &amp; ( n < k implies it . k = 0. X ) );
existence
 ex b1 being sequence of X st
for k being Element of NAT holds
( ( k <= n implies b1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) &amp; ( n < k implies b1 . k = 0. X ) )
proofend;
uniqueness
 for b1, b2 being sequence of X st ( for k being Element of NAT holds
( ( k <= n implies b1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) &amp; ( n < k implies b1 . k = 0. X ) ) ) &amp; ( for k being Element of NAT holds
( ( k <= n implies b2 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) &amp; ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def4 defines Alfa CLOPBAN4:def_4_:_
 for n being Element of NAT
for X being Complex_Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Alfa (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) &amp; ( n < k implies b5 . k = 0. X ) ) );

definition
let X be Complex_Banach_Algebra;
let z, w be Element of X;
let n be Element of NAT ;
func Conj (n,z,w) ->  sequence of X means :Def5: :: CLOPBAN4:def 5
 for k being Element of NAT holds
( ( k <= n implies it . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) &amp; ( n < k implies it . k = 0. X ) );
existence
 ex b1 being sequence of X st
for k being Element of NAT holds
( ( k <= n implies b1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) &amp; ( n < k implies b1 . k = 0. X ) )
proofend;
uniqueness
 for b1, b2 being sequence of X st ( for k being Element of NAT holds
( ( k <= n implies b1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) &amp; ( n < k implies b1 . k = 0. X ) ) ) &amp; ( for k being Element of NAT holds
( ( k <= n implies b2 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) &amp; ( n < k implies b2 . k = 0. X ) ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def5 defines Conj CLOPBAN4:def_5_:_
 for X being Complex_Banach_Algebra
for z, w being Element of X
for n being Element of NAT
for b5 being sequence of X holds
( b5 = Conj (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) &amp; ( n < k implies b5 . k = 0. X ) ) );

theorem Th13: :: CLOPBAN4:13
for X being Complex_Banach_Algebra
for z being Element of X
for n being Element of NAT holds
( (z ExpSeq) . (n + 1) = ((1r / (n + 1)) * z) * ((z ExpSeq) . n) &amp; (z ExpSeq) . 0 = 1. X &amp; ||.((z ExpSeq) . n).|| <= (||.z.|| rExpSeq) . n )
proofend;

theorem :: CLOPBAN4:14
for X being Complex_Banach_Algebra
for k being Element of NAT
for seq being sequence of X st 0 < k holds
(Shift seq) . k = seq . (k -' 1) by LOPBAN_4:15;

theorem Th15: :: CLOPBAN4:15
for X being Complex_Banach_Algebra
for k being Element of NAT
for seq being sequence of X holds (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k)
proofend;

theorem Th16: :: CLOPBAN4:16
for X being Complex_Banach_Algebra
for n being Element of NAT
for z, w being Element of X st z,w are_commutative holds
(z + w) #N n = (Partial_Sums (Expan (n,z,w))) . n
proofend;

theorem Th17: :: CLOPBAN4:17
for X being Complex_Banach_Algebra
for z, w being Element of X
for n being Element of NAT holds Expan_e (n,z,w) = (1r / (n !)) * (Expan (n,z,w))
proofend;

theorem Th18: :: CLOPBAN4:18
for X being Complex_Banach_Algebra
for n being Element of NAT
for z, w being Element of X st z,w are_commutative holds
(1r / (n !)) * ((z + w) #N n) = (Partial_Sums (Expan_e (n,z,w))) . n
proofend;

theorem Th19: :: CLOPBAN4:19
for X being Complex_Banach_Algebra holds
( (0. X) ExpSeq is norm_summable &amp; Sum ((0. X) ExpSeq) = 1. X )
proofend;

registration
let X be Complex_Banach_Algebra;
let z be Element of X;
clusterz ExpSeq  ->  norm_summable ;
coherence
 z ExpSeq is norm_summable
proofend;
end;

theorem Th20: :: CLOPBAN4:20
for X being Complex_Banach_Algebra
for z, w being Element of X holds
( (z ExpSeq) . 0 = 1. X &amp; (Expan (0,z,w)) . 0 = 1. X )
proofend;

theorem Th21: :: CLOPBAN4:21
for X being Complex_Banach_Algebra
for z, w being Element of X
for l, k being Element of NAT st l <= k holds
(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)
proofend;

theorem Th22: :: CLOPBAN4:22
for X being Complex_Banach_Algebra
for z, w being Element of X
for k being Element of NAT holds (Partial_Sums (Alfa ((k + 1),z,w))) . k = ((Partial_Sums (Alfa (k,z,w))) . k) + ((Partial_Sums (Expan_e ((k + 1),z,w))) . k)
proofend;

theorem Th23: :: CLOPBAN4:23
for X being Complex_Banach_Algebra
for z, w being Element of X
for k being Element of NAT holds (z ExpSeq) . k = (Expan_e (k,z,w)) . k
proofend;

theorem Th24: :: CLOPBAN4:24
for X being Complex_Banach_Algebra
for n being Element of NAT
for z, w being Element of X st z,w are_commutative holds
(Partial_Sums ((z + w) ExpSeq)) . n = (Partial_Sums (Alfa (n,z,w))) . n
proofend;

theorem Th25: :: CLOPBAN4:25
for X being Complex_Banach_Algebra
for k being Element of NAT
for z, w being Element of X st z,w are_commutative holds
(((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) = (Partial_Sums (Conj (k,z,w))) . k
proofend;

theorem Th26: :: CLOPBAN4:26
for X being Complex_Banach_Algebra
for z being Element of X
for n being Element of NAT holds 0 <= (||.z.|| rExpSeq) . n
proofend;

theorem Th27: :: CLOPBAN4:27
for X being Complex_Banach_Algebra
for z being Element of X
for k being Element of NAT holds
( ||.((Partial_Sums (z ExpSeq)) . k).|| <= (Partial_Sums (||.z.|| rExpSeq)) . k &amp; (Partial_Sums (||.z.|| rExpSeq)) . k <= Sum (||.z.|| rExpSeq) &amp; ||.((Partial_Sums (z ExpSeq)) . k).|| <= Sum (||.z.|| rExpSeq) )
proofend;

theorem Th28: :: CLOPBAN4:28
for X being Complex_Banach_Algebra
for z being Element of X holds 1 <= Sum (||.z.|| rExpSeq)
proofend;

theorem Th29: :: CLOPBAN4:29
for X being Complex_Banach_Algebra
for z being Element of X
for n, m being Element of NAT holds
( abs ((Partial_Sums (||.z.|| rExpSeq)) . n) = (Partial_Sums (||.z.|| rExpSeq)) . n &amp; ( n <= m implies abs (((Partial_Sums (||.z.|| rExpSeq)) . m) - ((Partial_Sums (||.z.|| rExpSeq)) . n)) = ((Partial_Sums (||.z.|| rExpSeq)) . m) - ((Partial_Sums (||.z.|| rExpSeq)) . n) ) )
proofend;

theorem Th30: :: CLOPBAN4:30
for X being Complex_Banach_Algebra
for z, w being Element of X
for k, n being Element of NAT holds abs ((Partial_Sums ||.(Conj (k,z,w)).||) . n) = (Partial_Sums ||.(Conj (k,z,w)).||) . n
proofend;

theorem Th31: :: CLOPBAN4:31
for X being Complex_Banach_Algebra
for z, w being Element of X
for p being real number st p > 0 holds
ex n being Element of NAT st
for k being Element of NAT st n <= k holds
abs ((Partial_Sums ||.(Conj (k,z,w)).||) . k) < p
proofend;

theorem Th32: :: CLOPBAN4:32
for X being Complex_Banach_Algebra
for z, w being Element of X
for seq being sequence of X st ( for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) holds
( seq is convergent &amp; lim seq = 0. X )
proofend;

Lm3:  for X being Complex_Banach_Algebra
for z, w being Element of X st z,w are_commutative holds
(Sum (z ExpSeq)) * (Sum (w ExpSeq)) = Sum ((z + w) ExpSeq)
proofend;

definition
let X be Complex_Banach_Algebra;
func exp_ X ->  Function of the carrier of X, the carrier of X means :Def6: :: CLOPBAN4:def 6
 for z being Element of X holds it . z = Sum (z ExpSeq);
existence
 ex b1 being Function of the carrier of X, the carrier of X st
for z being Element of X holds b1 . z = Sum (z ExpSeq)
proofend;
uniqueness
 for b1, b2 being Function of the carrier of X, the carrier of X st ( for z being Element of X holds b1 . z = Sum (z ExpSeq) ) &amp; ( for z being Element of X holds b2 . z = Sum (z ExpSeq) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def6 defines exp_ CLOPBAN4:def_6_:_
 for X being Complex_Banach_Algebra
for b2 being Function of the carrier of X, the carrier of X holds
( b2 = exp_ X iff for z being Element of X holds b2 . z = Sum (z ExpSeq) );

definition
let X be Complex_Banach_Algebra;
let z be Element of X;
func exp z ->  Element of X equals :: CLOPBAN4:def 7
(exp_ X) . z;
correctness
coherence
(exp_ X) . z is Element of X;
;
end;

:: deftheorem  defines exp CLOPBAN4:def_7_:_
 for X being Complex_Banach_Algebra
for z being Element of X holds exp z = (exp_ X) . z;

theorem :: CLOPBAN4:33
for X being Complex_Banach_Algebra
for z being Element of X holds exp z = Sum (z ExpSeq) by Def6;

theorem Th34: :: CLOPBAN4:34
for X being Complex_Banach_Algebra
for z1, z2 being Element of X st z1,z2 are_commutative holds
( exp (z1 + z2) = (exp z1) * (exp z2) &amp; exp (z2 + z1) = (exp z2) * (exp z1) &amp; exp (z1 + z2) = exp (z2 + z1) &amp; exp z1, exp z2 are_commutative )
proofend;

theorem :: CLOPBAN4:35
for X being Complex_Banach_Algebra
for z1, z2 being Element of X st z1,z2 are_commutative holds
z1 * (exp z2) = (exp z2) * z1
proofend;

theorem Th36: :: CLOPBAN4:36
for X being Complex_Banach_Algebra holds exp (0. X) = 1. X
proofend;

theorem Th37: :: CLOPBAN4:37
for X being Complex_Banach_Algebra
for z being Element of X holds
( (exp z) * (exp (- z)) = 1. X &amp; (exp (- z)) * (exp z) = 1. X )
proofend;

theorem :: CLOPBAN4:38
for X being Complex_Banach_Algebra
for z being Element of X holds
( exp z is invertible &amp; (exp z) " = exp (- z) &amp; exp (- z) is invertible &amp; (exp (- z)) " = exp z )
proofend;

theorem Th39: :: CLOPBAN4:39
for X being Complex_Banach_Algebra
for z being Element of X
for s, t being Complex holds s * z,t * z are_commutative
proofend;

theorem :: CLOPBAN4:40
for X being Complex_Banach_Algebra
for z being Element of X
for s, t being Complex holds
( (exp (s * z)) * (exp (t * z)) = exp ((s + t) * z) &amp; (exp (t * z)) * (exp (s * z)) = exp ((t + s) * z) &amp; exp ((s + t) * z) = exp ((t + s) * z) &amp; exp (s * z), exp (t * z) are_commutative )
proofend;