:: by Noboru Endou

::

:: Received April 6, 2004

:: Copyright (c) 2004-2021 Association of Mizar Users

Lm1: for X being Complex_Banach_Algebra

for z being Element of X

for n being 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 )

proof end;

Lm2: for X being Complex_Banach_Algebra

for n being 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 )

proof end;

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

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent & lim (seq1 - seq2) = 0. X holds

lim seq1 = lim seq2

proof end;

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 Nat holds s . n = z ) holds

lim s = z

for s being sequence of X

for z being Element of X st ( for n being Nat holds s . n = z ) holds

lim s = z

proof end;

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

for s, s9 being sequence of X st s is convergent & s9 is convergent holds

s * s9 is convergent

proof end;

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

for z being Element of X

for s being sequence of X st s is convergent holds

z * s is convergent

proof end;

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

for z being Element of X

for s being sequence of X st s is convergent holds

s * z is convergent

proof end;

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)

for z being Element of X

for s being sequence of X st s is convergent holds

lim (z * s) = z * (lim s)

proof end;

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

for z being Element of X

for s being sequence of X st s is convergent holds

lim (s * z) = (lim s) * z

proof end;

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)

for s, s9 being sequence of X st s is convergent & s9 is convergent holds

lim (s * s9) = (lim s) * (lim s9)

proof end;

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 )

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 )

proof end;

theorem Th10: :: CLOPBAN4:10

for X being Complex_Banach_Algebra

for k being Nat

for seq being sequence of X holds ||.((Partial_Sums seq) . k).|| <= (Partial_Sums ||.seq.||) . k

for k being Nat

for seq being sequence of X holds ||.((Partial_Sums seq) . k).|| <= (Partial_Sums ||.seq.||) . k

proof end;

theorem Th11: :: CLOPBAN4:11

for X being Complex_Banach_Algebra

for m being Nat

for seq1, seq2 being sequence of X st ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) holds

(Partial_Sums seq1) . m = (Partial_Sums seq2) . m

for m being Nat

for seq1, seq2 being sequence of X st ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) holds

(Partial_Sums seq1) . m = (Partial_Sums seq2) . m

proof end;

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 Nat holds ||.(seq . n).|| <= rseq . n ) & rseq is convergent & lim rseq = 0 holds

( seq is convergent & lim seq = 0. X )

for seq being sequence of X

for rseq being Real_Sequence st ( for n being Nat holds ||.(seq . n).|| <= rseq . n ) & rseq is convergent & lim rseq = 0 holds

( seq is convergent & lim seq = 0. X )

proof end;

definition

let X be Complex_Banach_Algebra;

let z be Element of X;

ex b_{1} being sequence of X st

for n being Nat holds b_{1} . n = (1r / (n !)) * (z #N n)

for b_{1}, b_{2} being sequence of X st ( for n being Nat holds b_{1} . n = (1r / (n !)) * (z #N n) ) & ( for n being Nat holds b_{2} . n = (1r / (n !)) * (z #N n) ) holds

b_{1} = b_{2}

end;
let z be Element of X;

func z ExpSeq -> sequence of X means :Def1: :: CLOPBAN4:def 1

for n being Nat holds it . n = (1r / (n !)) * (z #N n);

existence for n being Nat holds it . n = (1r / (n !)) * (z #N n);

ex b

for n being Nat holds b

proof end;

uniqueness for b

b

proof end;

:: deftheorem Def1 defines ExpSeq CLOPBAN4:def 1 :

for X being Complex_Banach_Algebra

for z being Element of X

for b_{3} being sequence of X holds

( b_{3} = z ExpSeq iff for n being Nat holds b_{3} . n = (1r / (n !)) * (z #N n) );

for X being Complex_Banach_Algebra

for z being Element of X

for b

( b

definition

let n be Nat;

let X be Complex_Banach_Algebra;

let z, w be Element of X;

ex b_{1} being sequence of X st

for k being Nat holds

( ( k <= n implies b_{1} . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{1} . k = 0. X ) )

for b_{1}, b_{2} being sequence of X st ( for k being Nat holds

( ( k <= n implies b_{1} . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{1} . k = 0. X ) ) ) & ( for k being Nat holds

( ( k <= n implies b_{2} . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{2} . k = 0. X ) ) ) holds

b_{1} = b_{2}

end;
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 Nat holds

( ( k <= n implies it . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies it . k = 0. X ) );

existence for k being Nat holds

( ( k <= n implies it . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies it . k = 0. X ) );

ex b

for k being Nat holds

( ( k <= n implies b

proof end;

uniqueness for b

( ( k <= n implies b

( ( k <= n implies b

b

proof end;

:: deftheorem Def2 defines Expan CLOPBAN4:def 2 :

for n being Nat

for X being Complex_Banach_Algebra

for z, w being Element of X

for b_{5} being sequence of X holds

( b_{5} = Expan (n,z,w) iff for k being Nat holds

( ( k <= n implies b_{5} . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{5} . k = 0. X ) ) );

for n being Nat

for X being Complex_Banach_Algebra

for z, w being Element of X

for b

( b

( ( k <= n implies b

definition

let n be Nat;

let X be Complex_Banach_Algebra;

let z, w be Element of X;

ex b_{1} being sequence of X st

for k being Nat holds

( ( k <= n implies b_{1} . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{1} . k = 0. X ) )

for b_{1}, b_{2} being sequence of X st ( for k being Nat holds

( ( k <= n implies b_{1} . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{1} . k = 0. X ) ) ) & ( for k being Nat holds

( ( k <= n implies b_{2} . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{2} . k = 0. X ) ) ) holds

b_{1} = b_{2}

end;
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 Nat holds

( ( k <= n implies it . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies it . k = 0. X ) );

existence for k being Nat holds

( ( k <= n implies it . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies it . k = 0. X ) );

ex b

for k being Nat holds

( ( k <= n implies b

proof end;

uniqueness for b

( ( k <= n implies b

( ( k <= n implies b

b

proof end;

:: deftheorem Def3 defines Expan_e CLOPBAN4:def 3 :

for n being Nat

for X being Complex_Banach_Algebra

for z, w being Element of X

for b_{5} being sequence of X holds

( b_{5} = Expan_e (n,z,w) iff for k being Nat holds

( ( k <= n implies b_{5} . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b_{5} . k = 0. X ) ) );

for n being Nat

for X being Complex_Banach_Algebra

for z, w being Element of X

for b

( b

( ( k <= n implies b

definition

let n be Nat;

let X be Complex_Banach_Algebra;

let z, w be Element of X;

ex b_{1} being sequence of X st

for k being Nat holds

( ( k <= n implies b_{1} . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b_{1} . k = 0. X ) )

for b_{1}, b_{2} being sequence of X st ( for k being Nat holds

( ( k <= n implies b_{1} . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b_{1} . k = 0. X ) ) ) & ( for k being Nat holds

( ( k <= n implies b_{2} . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b_{2} . k = 0. X ) ) ) holds

b_{1} = b_{2}

end;
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 Nat holds

( ( k <= n implies it . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies it . k = 0. X ) );

existence for k being Nat holds

( ( k <= n implies it . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies it . k = 0. X ) );

ex b

for k being Nat holds

( ( k <= n implies b

proof end;

uniqueness for b

( ( k <= n implies b

( ( k <= n implies b

b

proof end;

:: deftheorem Def4 defines Alfa CLOPBAN4:def 4 :

for n being Nat

for X being Complex_Banach_Algebra

for z, w being Element of X

for b_{5} being sequence of X holds

( b_{5} = Alfa (n,z,w) iff for k being Nat holds

( ( k <= n implies b_{5} . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b_{5} . k = 0. X ) ) );

for n being Nat

for X being Complex_Banach_Algebra

for z, w being Element of X

for b

( b

( ( k <= n implies b

definition

let X be Complex_Banach_Algebra;

let z, w be Element of X;

let n be Nat;

ex b_{1} being sequence of X st

for k being Nat holds

( ( k <= n implies b_{1} . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b_{1} . k = 0. X ) )

for b_{1}, b_{2} being sequence of X st ( for k being Nat holds

( ( k <= n implies b_{1} . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b_{1} . k = 0. X ) ) ) & ( for k being Nat holds

( ( k <= n implies b_{2} . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b_{2} . k = 0. X ) ) ) holds

b_{1} = b_{2}

end;
let z, w be Element of X;

let n be Nat;

func Conj (n,z,w) -> sequence of X means :Def5: :: CLOPBAN4:def 5

for k being Nat holds

( ( k <= n implies it . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies it . k = 0. X ) );

existence for k being Nat holds

( ( k <= n implies it . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies it . k = 0. X ) );

ex b

for k being Nat holds

( ( k <= n implies b

proof end;

uniqueness for b

( ( k <= n implies b

( ( k <= n implies b

b

proof end;

:: deftheorem Def5 defines Conj CLOPBAN4:def 5 :

for X being Complex_Banach_Algebra

for z, w being Element of X

for n being Nat

for b_{5} being sequence of X holds

( b_{5} = Conj (n,z,w) iff for k being Nat holds

( ( k <= n implies b_{5} . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b_{5} . k = 0. X ) ) );

for X being Complex_Banach_Algebra

for z, w being Element of X

for n being Nat

for b

( b

( ( k <= n implies b

theorem Th13: :: CLOPBAN4:13

for X being Complex_Banach_Algebra

for z being Element of X

for n being Nat holds

( (z ExpSeq) . (n + 1) = ((1r / (n + 1)) * z) * ((z ExpSeq) . n) & (z ExpSeq) . 0 = 1. X & ||.((z ExpSeq) . n).|| <= (||.z.|| rExpSeq) . n )

for z being Element of X

for n being Nat holds

( (z ExpSeq) . (n + 1) = ((1r / (n + 1)) * z) * ((z ExpSeq) . n) & (z ExpSeq) . 0 = 1. X & ||.((z ExpSeq) . n).|| <= (||.z.|| rExpSeq) . n )

proof end;

theorem :: CLOPBAN4:14

theorem Th15: :: CLOPBAN4:15

for X being Complex_Banach_Algebra

for k being Nat

for seq being sequence of X holds (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k)

for k being Nat

for seq being sequence of X holds (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k)

proof end;

theorem Th16: :: CLOPBAN4:16

for X being Complex_Banach_Algebra

for n being 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

for n being 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

proof end;

theorem Th17: :: CLOPBAN4:17

for X being Complex_Banach_Algebra

for w, z being Element of X

for n being Nat holds Expan_e (n,z,w) = (1r / (n !)) * (Expan (n,z,w))

for w, z being Element of X

for n being Nat holds Expan_e (n,z,w) = (1r / (n !)) * (Expan (n,z,w))

proof end;

theorem Th18: :: CLOPBAN4:18

for X being Complex_Banach_Algebra

for n being 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

for n being 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

proof end;

theorem Th19: :: CLOPBAN4:19

for X being Complex_Banach_Algebra holds

( (0. X) ExpSeq is norm_summable & Sum ((0. X) ExpSeq) = 1. X )

( (0. X) ExpSeq is norm_summable & Sum ((0. X) ExpSeq) = 1. X )

proof end;

registration

let X be Complex_Banach_Algebra;

let z be Element of X;

coherence

z ExpSeq is norm_summable

end;
let z be Element of X;

coherence

z ExpSeq is norm_summable

proof end;

theorem Th20: :: CLOPBAN4:20

for X being Complex_Banach_Algebra

for w, z being Element of X holds

( (z ExpSeq) . 0 = 1. X & (Expan (0,z,w)) . 0 = 1. X )

for w, z being Element of X holds

( (z ExpSeq) . 0 = 1. X & (Expan (0,z,w)) . 0 = 1. X )

proof end;

theorem Th21: :: CLOPBAN4:21

for X being Complex_Banach_Algebra

for w, z being Element of X

for k, l being Nat st l <= k holds

(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)

for w, z being Element of X

for k, l being Nat st l <= k holds

(Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l)

proof end;

theorem Th22: :: CLOPBAN4:22

for X being Complex_Banach_Algebra

for w, z being Element of X

for k being 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)

for w, z being Element of X

for k being 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)

proof end;

theorem Th23: :: CLOPBAN4:23

for X being Complex_Banach_Algebra

for w, z being Element of X

for k being Nat holds (z ExpSeq) . k = (Expan_e (k,z,w)) . k

for w, z being Element of X

for k being Nat holds (z ExpSeq) . k = (Expan_e (k,z,w)) . k

proof end;

theorem Th24: :: CLOPBAN4:24

for X being Complex_Banach_Algebra

for n being 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

for n being 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

proof end;

theorem Th25: :: CLOPBAN4:25

for X being Complex_Banach_Algebra

for k being 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

for k being 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

proof end;

theorem Th26: :: CLOPBAN4:26

for X being Complex_Banach_Algebra

for z being Element of X

for n being Nat holds 0 <= (||.z.|| rExpSeq) . n

for z being Element of X

for n being Nat holds 0 <= (||.z.|| rExpSeq) . n

proof end;

theorem Th27: :: CLOPBAN4:27

for X being Complex_Banach_Algebra

for z being Element of X

for k being Nat holds

( ||.((Partial_Sums (z ExpSeq)) . k).|| <= (Partial_Sums (||.z.|| rExpSeq)) . k & (Partial_Sums (||.z.|| rExpSeq)) . k <= Sum (||.z.|| rExpSeq) & ||.((Partial_Sums (z ExpSeq)) . k).|| <= Sum (||.z.|| rExpSeq) )

for z being Element of X

for k being Nat holds

( ||.((Partial_Sums (z ExpSeq)) . k).|| <= (Partial_Sums (||.z.|| rExpSeq)) . k & (Partial_Sums (||.z.|| rExpSeq)) . k <= Sum (||.z.|| rExpSeq) & ||.((Partial_Sums (z ExpSeq)) . k).|| <= Sum (||.z.|| rExpSeq) )

proof end;

theorem Th29: :: CLOPBAN4:29

for X being Complex_Banach_Algebra

for z being Element of X

for m, n being Nat holds

( |.((Partial_Sums (||.z.|| rExpSeq)) . n).| = (Partial_Sums (||.z.|| rExpSeq)) . n & ( n <= m implies |.(((Partial_Sums (||.z.|| rExpSeq)) . m) - ((Partial_Sums (||.z.|| rExpSeq)) . n)).| = ((Partial_Sums (||.z.|| rExpSeq)) . m) - ((Partial_Sums (||.z.|| rExpSeq)) . n) ) )

for z being Element of X

for m, n being Nat holds

( |.((Partial_Sums (||.z.|| rExpSeq)) . n).| = (Partial_Sums (||.z.|| rExpSeq)) . n & ( n <= m implies |.(((Partial_Sums (||.z.|| rExpSeq)) . m) - ((Partial_Sums (||.z.|| rExpSeq)) . n)).| = ((Partial_Sums (||.z.|| rExpSeq)) . m) - ((Partial_Sums (||.z.|| rExpSeq)) . n) ) )

proof end;

theorem Th30: :: CLOPBAN4:30

for X being Complex_Banach_Algebra

for w, z being Element of X

for k, n being Nat holds |.((Partial_Sums ||.(Conj (k,z,w)).||) . n).| = (Partial_Sums ||.(Conj (k,z,w)).||) . n

for w, z being Element of X

for k, n being Nat holds |.((Partial_Sums ||.(Conj (k,z,w)).||) . n).| = (Partial_Sums ||.(Conj (k,z,w)).||) . n

proof end;

theorem Th31: :: CLOPBAN4:31

for X being Complex_Banach_Algebra

for w, z being Element of X

for p being Real st p > 0 holds

ex n being Nat st

for k being Nat st n <= k holds

|.((Partial_Sums ||.(Conj (k,z,w)).||) . k).| < p

for w, z being Element of X

for p being Real st p > 0 holds

ex n being Nat st

for k being Nat st n <= k holds

|.((Partial_Sums ||.(Conj (k,z,w)).||) . k).| < p

proof end;

theorem Th32: :: CLOPBAN4:32

for X being Complex_Banach_Algebra

for w, z being Element of X

for seq being sequence of X st ( for k being Nat holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) holds

( seq is convergent & lim seq = 0. X )

for w, z being Element of X

for seq being sequence of X st ( for k being Nat holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) holds

( seq is convergent & lim seq = 0. X )

proof end;

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)

proof end;

definition

let X be Complex_Banach_Algebra;

ex b_{1} being Function of the carrier of X, the carrier of X st

for z being Element of X holds b_{1} . z = Sum (z ExpSeq)

for b_{1}, b_{2} being Function of the carrier of X, the carrier of X st ( for z being Element of X holds b_{1} . z = Sum (z ExpSeq) ) & ( for z being Element of X holds b_{2} . z = Sum (z ExpSeq) ) holds

b_{1} = b_{2}

end;
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 for z being Element of X holds it . z = Sum (z ExpSeq);

ex b

for z being Element of X holds b

proof end;

uniqueness for b

b

proof end;

:: deftheorem Def6 defines exp_ CLOPBAN4:def 6 :

for X being Complex_Banach_Algebra

for b_{2} being Function of the carrier of X, the carrier of X holds

( b_{2} = exp_ X iff for z being Element of X holds b_{2} . z = Sum (z ExpSeq) );

for X being Complex_Banach_Algebra

for b

( b

definition

let X be Complex_Banach_Algebra;

let z be Element of X;

correctness

coherence

(exp_ X) . z is Element of X;

;

end;
let z be Element of X;

correctness

coherence

(exp_ X) . z is Element of X;

;

:: deftheorem defines exp CLOPBAN4:def 7 :

for X being Complex_Banach_Algebra

for z being Element of X holds exp z = (exp_ X) . z;

for X being Complex_Banach_Algebra

for z being Element of X holds exp z = (exp_ X) . z;

theorem :: CLOPBAN4:33

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) & exp (z2 + z1) = (exp z2) * (exp z1) & exp (z1 + z2) = exp (z2 + z1) & exp z1, exp z2 are_commutative )

for z1, z2 being Element of X st z1,z2 are_commutative holds

( exp (z1 + z2) = (exp z1) * (exp z2) & exp (z2 + z1) = (exp z2) * (exp z1) & exp (z1 + z2) = exp (z2 + z1) & exp z1, exp z2 are_commutative )

proof end;

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

for z1, z2 being Element of X st z1,z2 are_commutative holds

z1 * (exp z2) = (exp z2) * z1

proof end;

theorem Th37: :: CLOPBAN4:37

for X being Complex_Banach_Algebra

for z being Element of X holds

( (exp z) * (exp (- z)) = 1. X & (exp (- z)) * (exp z) = 1. X )

for z being Element of X holds

( (exp z) * (exp (- z)) = 1. X & (exp (- z)) * (exp z) = 1. X )

proof end;

theorem :: CLOPBAN4:38

for X being Complex_Banach_Algebra

for z being Element of X holds

( exp z is invertible & (exp z) " = exp (- z) & exp (- z) is invertible & (exp (- z)) " = exp z )

for z being Element of X holds

( exp z is invertible & (exp z) " = exp (- z) & exp (- z) is invertible & (exp (- z)) " = exp z )

proof end;

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

for z being Element of X

for s, t being Complex holds s * z,t * z are_commutative

proof end;

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) & (exp (t * z)) * (exp (s * z)) = exp ((t + s) * z) & exp ((s + t) * z) = exp ((t + s) * z) & exp (s * z), exp (t * z) are_commutative )

for z being Element of X

for s, t being Complex holds

( (exp (s * z)) * (exp (t * z)) = exp ((s + t) * z) & (exp (t * z)) * (exp (s * z)) = exp ((t + s) * z) & exp ((s + t) * z) = exp ((t + s) * z) & exp (s * z), exp (t * z) are_commutative )

proof end;