:: The Exponential Function on {B}anach Algebra
:: by Yasunari Shidama
::
:: Received February 13, 2004
:: Copyright (c) 2004 Association of Mizar Users
:: deftheorem Def1 defines are_commutative LOPBAN_4:def 1 :
Lm1:
for X being 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 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: :: LOPBAN_4:1
theorem Th2: :: LOPBAN_4:2
theorem Th3: :: LOPBAN_4:3
theorem Th4: :: LOPBAN_4:4
theorem Th5: :: LOPBAN_4:5
theorem Th6: :: LOPBAN_4:6
theorem Th7: :: LOPBAN_4:7
theorem Th8: :: LOPBAN_4:8
theorem Th9: :: LOPBAN_4:9
theorem Th10: :: LOPBAN_4:10
theorem Th11: :: LOPBAN_4:11
theorem Th12: :: LOPBAN_4:12
:: deftheorem Def2 defines rExpSeq LOPBAN_4:def 2 :
theorem Th13: :: LOPBAN_4:13
:: deftheorem Def3 defines Coef LOPBAN_4:def 3 :
:: deftheorem Def4 defines Coef_e LOPBAN_4:def 4 :
:: deftheorem Def5 defines Shift LOPBAN_4:def 5 :
:: deftheorem Def6 defines Expan LOPBAN_4:def 6 :
:: deftheorem Def7 defines Expan_e LOPBAN_4:def 7 :
:: deftheorem Def8 defines Alfa LOPBAN_4:def 8 :
:: deftheorem Def9 defines Conj LOPBAN_4:def 9 :
theorem Th14: :: LOPBAN_4:14
theorem Th15: :: LOPBAN_4:15
theorem Th16: :: LOPBAN_4:16
theorem Th17: :: LOPBAN_4:17
theorem Th18: :: LOPBAN_4:18
theorem Th19: :: LOPBAN_4:19
theorem Th20: :: LOPBAN_4:20
theorem Th21: :: LOPBAN_4:21
theorem Th22: :: LOPBAN_4:22
theorem Th23: :: LOPBAN_4:23
theorem Th24: :: LOPBAN_4:24
theorem Th25: :: LOPBAN_4:25
theorem Th26: :: LOPBAN_4:26
theorem Th27: :: LOPBAN_4:27
theorem Th28: :: LOPBAN_4:28
theorem Th29: :: LOPBAN_4:29
theorem Th30: :: LOPBAN_4:30
theorem Th31: :: LOPBAN_4:31
theorem Th32: :: LOPBAN_4:32
theorem Th33: :: LOPBAN_4:33
Lm3:
for X being Banach_Algebra
for z, w being Element of X st z,w are_commutative holds
(Sum (z rExpSeq )) * (Sum (w rExpSeq )) = Sum ((z + w) rExpSeq )
:: deftheorem Def10 defines exp_ LOPBAN_4:def 10 :
:: deftheorem defines exp LOPBAN_4:def 11 :
theorem :: LOPBAN_4:34
theorem Th35: :: LOPBAN_4:35
theorem :: LOPBAN_4:36
theorem Th37: :: LOPBAN_4:37
theorem Th38: :: LOPBAN_4:38
theorem :: LOPBAN_4:39
theorem Th40: :: LOPBAN_4:40
theorem :: LOPBAN_4:41