:: Exponential Function on Complex {B}anach Algebra
:: by Noboru Endou
::
:: Received April 6, 2004
:: Copyright (c) 2004 Association of Mizar Users
definition
canceled;
end;
:: deftheorem CLOPBAN4:def 1 :
canceled;
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 )
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 )
theorem Th1: :: CLOPBAN4:1
theorem Th2: :: CLOPBAN4:2
theorem Th3: :: CLOPBAN4:3
theorem Th4: :: CLOPBAN4:4
theorem Th5: :: CLOPBAN4:5
theorem Th6: :: CLOPBAN4:6
theorem Th7: :: CLOPBAN4:7
theorem Th8: :: CLOPBAN4:8
theorem Th9: :: CLOPBAN4:9
theorem Th10: :: CLOPBAN4:10
theorem Th11: :: CLOPBAN4:11
theorem Th12: :: CLOPBAN4:12
:: deftheorem Def2 defines ExpSeq CLOPBAN4:def 2 :
:: deftheorem CLOPBAN4:def 3 :
canceled;
:: deftheorem Def4 defines Expan CLOPBAN4:def 4 :
:: deftheorem Def5 defines Expan_e CLOPBAN4:def 5 :
:: deftheorem Def6 defines Alfa CLOPBAN4:def 6 :
:: deftheorem Def7 defines Conj CLOPBAN4:def 7 :
theorem Th13: :: CLOPBAN4:13
theorem :: CLOPBAN4:14
theorem Th15: :: CLOPBAN4:15
theorem Th16: :: CLOPBAN4:16
theorem Th17: :: CLOPBAN4:17
theorem Th18: :: CLOPBAN4:18
theorem Th19: :: CLOPBAN4:19
theorem Th20: :: CLOPBAN4:20
theorem Th21: :: CLOPBAN4:21
theorem Th22: :: CLOPBAN4:22
theorem Th23: :: CLOPBAN4:23
theorem Th24: :: CLOPBAN4:24
theorem Th25: :: CLOPBAN4:25
theorem Th26: :: CLOPBAN4:26
theorem Th27: :: CLOPBAN4:27
theorem Th28: :: CLOPBAN4:28
theorem Th29: :: CLOPBAN4:29
theorem Th30: :: CLOPBAN4:30
theorem Th31: :: CLOPBAN4:31
theorem Th32: :: CLOPBAN4:32
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 )
:: deftheorem Def8 defines exp_ CLOPBAN4:def 8 :
:: deftheorem defines exp CLOPBAN4:def 9 :
theorem :: CLOPBAN4:33
theorem Th34: :: CLOPBAN4:34
theorem :: CLOPBAN4:35
theorem Th36: :: CLOPBAN4:36
theorem Th37: :: CLOPBAN4:37
theorem :: CLOPBAN4:38
theorem Th39: :: CLOPBAN4:39
theorem :: CLOPBAN4:40