:: CLVECT_1 semantic presentation
:: deftheorem Def1 defines * CLVECT_1:def 1 :
E1:
now
take c
1 =
{0};
reconsider c
2 = 0 as
Element of c
1 by TARSKI:def 1;
take c
3 = c
2;
deffunc H
1(
Element of c
1,
Element of c
1)
-> Element of c
1 = c
3;
consider c
4 being
BinOp of c
1 such that E2:
for b
1, b
2 being
Element of c
1 holds c
4 . b
1,b
2 = H
1(b
1,b
2)
from BINOP_1:sch 4();
reconsider c
5 =
[:COMPLEX ,c1:] --> c
3 as
Function of
[:COMPLEX ,c1:],c
1 by FUNCOP_1:57;
E3:
for b
1 being
Element of
COMPLEX for b
2 being
Element of c
1 holds c
5 . [b1,b2] = c
3
take c
6 = c
4;
take c
7 = c
5;
set c
8 =
CLSStruct(# c
1,c
3,c
6,c
7 #);
thus
for b
1, b
2 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds b
1 + b
2 = b
2 + b
1
proof
let c
9, c
10 be
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #);
E4:
( c
9 + c
10 = c
6 . c
9,c
10 & c
10 + c
9 = c
6 . c
10,c
9 )
;
reconsider c
11 = c
9, c
12 = c
10 as
Element of c
1 ;
( c
9 + c
10 = H
1(c
11,c
12) & c
10 + c
9 = H
1(c
12,c
11) )
by E2, E4;
hence
c
9 + c
10 = c
10 + c
9
;
end;
thus
for b
1, b
2, b
3 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds
(b1 + b2) + b
3 = b
1 + (b2 + b3)
proof
let c
9, c
10, c
11 be
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #);
E4:
(
(c9 + c10) + c
11 = c
6 . (c9 + c10),c
11 & c
9 + (c10 + c11) = c
6 . c
9,
(c10 + c11) )
;
reconsider c
12 = c
9, c
13 = c
10, c
14 = c
11 as
Element of c
1 ;
(
(c9 + c10) + c
11 = H
1(H
1(c
12,c
13),c
14) & c
9 + (c10 + c11) = H
1(c
12,H
1(c
13,c
14)) )
by E2, E4;
hence
(c9 + c10) + c
11 = c
9 + (c10 + c11)
;
end;
thus
for b
1 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds b
1 + (0. CLSStruct(# c1,c3,c6,c7 #)) = b
1
proof
let c
9 be
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #);
reconsider c
10 = c
9 as
Element of c
1 ;
c
9 + (0. CLSStruct(# c1,c3,c6,c7 #)) =
c
6 . [c9,(0. CLSStruct(# c1,c3,c6,c7 #))]
.=
c
6 . c
9,
(0. CLSStruct(# c1,c3,c6,c7 #))
.=
H
1(c
10,c
3)
by E2
;
hence
c
9 + (0. CLSStruct(# c1,c3,c6,c7 #)) = c
9
by TARSKI:def 1;
end;
thus
for b
1 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds
ex b
2 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) st b
1 + b
2 = 0. CLSStruct(# c
1,c
3,c
6,c
7 #)
proof
let c
9 be
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #);
reconsider c
10 = c
3 as
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) ;
take
c
10
;
thus c
9 + c
10 =
c
6 . [c9,c10]
.=
c
6 . c
9,c
10
.=
the
Zero of
CLSStruct(# c
1,c
3,c
6,c
7 #)
by E2
.=
0. CLSStruct(# c
1,c
3,c
6,c
7 #)
;
end;
thus
for b
1 being
Complexfor b
2, b
3 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds b
1 * (b2 + b3) = (b1 * b2) + (b1 * b3)
proof
let c
9 be
Complex;
let c
10, c
11 be
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #);
reconsider c
12 = c
10, c
13 = c
11 as
Element of c
1 ;
(c9 * c10) + (c9 * c11) =
c
6 . [(c9 * c10),(c9 * c11)]
.=
c
6 . (c9 * c10),
(c9 * c11)
.=
H
1(c
3,c
3)
by E2
;
hence
c
9 * (c10 + c11) = (c9 * c10) + (c9 * c11)
by E3;
end;
thus
for b
1, b
2 being
Complexfor b
3 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds
(b1 + b2) * b
3 = (b1 * b3) + (b2 * b3)
proof
let c
9, c
10 be
Complex;
let c
11 be
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #);
set c
12 = c
9 + c
10;
reconsider c
13 = c
11 as
Element of c
1 ;
E4:
(c9 + c10) * c
11 =
c
7 . [(c9 + c10),c11]
.=
c
3
by E3
;
(c9 * c11) + (c10 * c11) =
c
6 . [(c9 * c11),(c10 * c11)]
.=
c
6 . (c9 * c11),
(c10 * c11)
.=
H
1(c
3,c
3)
by E2
;
hence
(c9 + c10) * c
11 = (c9 * c11) + (c10 * c11)
by E4;
end;
thus
for b
1, b
2 being
Complexfor b
3 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds
(b1 * b2) * b
3 = b
1 * (b2 * b3)
proof
let c
9, c
10 be
Complex;
let c
11 be
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #);
set c
12 = c
9 * c
10;
reconsider c
13 = c
11 as
Element of c
1 ;
E4:
(c9 * c10) * c
11 =
c
7 . [(c9 * c10),c11]
.=
c
3
by E3
;
c
9 * (c10 * c11) =
c
7 . [c9,(c10 * c11)]
.=
c
3
by E3
;
hence
(c9 * c10) * c
11 = c
9 * (c10 * c11)
by E4;
end;
thus
for b
1 being
VECTOR of
CLSStruct(# c
1,c
3,c
6,c
7 #) holds
1r * b
1 = b
1
end;
:: deftheorem Def2 defines ComplexLinearSpace-like CLVECT_1:def 2 :
theorem Th1: :: CLVECT_1:1
for b
1 being non
empty CLSStruct holds
( ( for b
2, b
3 being
VECTOR of b
1 holds b
2 + b
3 = b
3 + b
2 ) & ( for b
2, b
3, b
4 being
VECTOR of b
1 holds
(b2 + b3) + b
4 = b
2 + (b3 + b4) ) & ( for b
2 being
VECTOR of b
1 holds b
2 + (0. b1) = b
2 ) & ( for b
2 being
VECTOR of b
1 holds
ex b
3 being
VECTOR of b
1 st b
2 + b
3 = 0. b
1 ) & ( for b
2 being
Complexfor b
3, b
4 being
VECTOR of b
1 holds b
2 * (b3 + b4) = (b2 * b3) + (b2 * b4) ) & ( for b
2, b
3 being
Complexfor b
4 being
VECTOR of b
1 holds
(b2 + b3) * b
4 = (b2 * b4) + (b3 * b4) ) & ( for b
2, b
3 being
Complexfor b
4 being
VECTOR of b
1 holds
(b2 * b3) * b
4 = b
2 * (b3 * b4) ) & ( for b
2 being
VECTOR of b
1 holds
1r * b
2 = b
2 ) implies b
1 is
ComplexLinearSpace )
by Def2, RLVECT_1:def 5, RLVECT_1:def 6, RLVECT_1:def 7, RLVECT_1:def 8;
theorem Th2: :: CLVECT_1:2
theorem Th3: :: CLVECT_1:3
theorem Th4: :: CLVECT_1:4
theorem Th5: :: CLVECT_1:5
theorem Th6: :: CLVECT_1:6
theorem Th7: :: CLVECT_1:7
theorem Th8: :: CLVECT_1:8
theorem Th9: :: CLVECT_1:9
theorem Th10: :: CLVECT_1:10
theorem Th11: :: CLVECT_1:11
theorem Th12: :: CLVECT_1:12
theorem Th13: :: CLVECT_1:13
Lemma11:
for b1 being non empty LoopStr holds Sum (<*> the carrier of b1) = 0. b1
Lemma12:
for b1 being non empty LoopStr
for b2 being FinSequence of the carrier of b1 holds
( len b2 = 0 implies Sum b2 = 0. b1 )
theorem Th14: :: CLVECT_1:14
theorem Th15: :: CLVECT_1:15
theorem Th16: :: CLVECT_1:16
theorem Th17: :: CLVECT_1:17
Lemma13:
1r + 1r = [*2,0*]
theorem Th18: :: CLVECT_1:18
theorem Th19: :: CLVECT_1:19
theorem Th20: :: CLVECT_1:20
:: deftheorem Def3 defines lineary-closed CLVECT_1:def 3 :
theorem Th21: :: CLVECT_1:21
theorem Th22: :: CLVECT_1:22
theorem Th23: :: CLVECT_1:23
theorem Th24: :: CLVECT_1:24
theorem Th25: :: CLVECT_1:25
theorem Th26: :: CLVECT_1:26
theorem Th27: :: CLVECT_1:27
:: deftheorem Def4 defines Subspace CLVECT_1:def 4 :
theorem Th28: :: CLVECT_1:28
theorem Th29: :: CLVECT_1:29
theorem Th30: :: CLVECT_1:30
theorem Th31: :: CLVECT_1:31
theorem Th32: :: CLVECT_1:32
theorem Th33: :: CLVECT_1:33
theorem Th34: :: CLVECT_1:34
theorem Th35: :: CLVECT_1:35
theorem Th36: :: CLVECT_1:36
Lemma27:
for b1 being ComplexLinearSpace
for b2 being Subset of b1
for b3 being Subspace of b1 holds
( the carrier of b3 = b2 implies b2 is lineary-closed )
theorem Th37: :: CLVECT_1:37
theorem Th38: :: CLVECT_1:38
theorem Th39: :: CLVECT_1:39
theorem Th40: :: CLVECT_1:40
theorem Th41: :: CLVECT_1:41
theorem Th42: :: CLVECT_1:42
theorem Th43: :: CLVECT_1:43
theorem Th44: :: CLVECT_1:44
theorem Th45: :: CLVECT_1:45
theorem Th46: :: CLVECT_1:46
theorem Th47: :: CLVECT_1:47
theorem Th48: :: CLVECT_1:48
theorem Th49: :: CLVECT_1:49
theorem Th50: :: CLVECT_1:50
theorem Th51: :: CLVECT_1:51
theorem Th52: :: CLVECT_1:52
theorem Th53: :: CLVECT_1:53
theorem Th54: :: CLVECT_1:54
theorem Th55: :: CLVECT_1:55
:: deftheorem Def5 defines (0). CLVECT_1:def 5 :
:: deftheorem Def6 defines (Omega). CLVECT_1:def 6 :
theorem Th56: :: CLVECT_1:56
theorem Th57: :: CLVECT_1:57
theorem Th58: :: CLVECT_1:58
theorem Th59: :: CLVECT_1:59
theorem Th60: :: CLVECT_1:60
theorem Th61: :: CLVECT_1:61
:: deftheorem Def7 defines + CLVECT_1:def 7 :
Lemma44:
for b1 being ComplexLinearSpace
for b2 being Subspace of b1 holds (0. b1) + b2 = the carrier of b2
:: deftheorem Def8 defines Coset CLVECT_1:def 8 :
theorem Th62: :: CLVECT_1:62
theorem Th63: :: CLVECT_1:63
theorem Th64: :: CLVECT_1:64
theorem Th65: :: CLVECT_1:65
Lemma49:
for b1 being ComplexLinearSpace
for b2 being VECTOR of b1
for b3 being Subspace of b1 holds
( b2 in b3 iff b2 + b3 = the carrier of b3 )
theorem Th66: :: CLVECT_1:66
theorem Th67: :: CLVECT_1:67
theorem Th68: :: CLVECT_1:68
theorem Th69: :: CLVECT_1:69
theorem Th70: :: CLVECT_1:70
theorem Th71: :: CLVECT_1:71
theorem Th72: :: CLVECT_1:72
theorem Th73: :: CLVECT_1:73
theorem Th74: :: CLVECT_1:74
theorem Th75: :: CLVECT_1:75
theorem Th76: :: CLVECT_1:76
theorem Th77: :: CLVECT_1:77
theorem Th78: :: CLVECT_1:78
theorem Th79: :: CLVECT_1:79
theorem Th80: :: CLVECT_1:80
theorem Th81: :: CLVECT_1:81
theorem Th82: :: CLVECT_1:82
theorem Th83: :: CLVECT_1:83
theorem Th84: :: CLVECT_1:84
theorem Th85: :: CLVECT_1:85
theorem Th86: :: CLVECT_1:86
theorem Th87: :: CLVECT_1:87
theorem Th88: :: CLVECT_1:88
theorem Th89: :: CLVECT_1:89
theorem Th90: :: CLVECT_1:90
theorem Th91: :: CLVECT_1:91
theorem Th92: :: CLVECT_1:92
theorem Th93: :: CLVECT_1:93
theorem Th94: :: CLVECT_1:94
theorem Th95: :: CLVECT_1:95
theorem Th96: :: CLVECT_1:96
theorem Th97: :: CLVECT_1:97
theorem Th98: :: CLVECT_1:98
theorem Th99: :: CLVECT_1:99
theorem Th100: :: CLVECT_1:100
theorem Th101: :: CLVECT_1:101
theorem Th102: :: CLVECT_1:102
deffunc H1( CNORMSTR ) -> Element of the carrier of a1 = 0. a1;
:: deftheorem Def9 defines ||. CLVECT_1:def 9 :
consider c1 being ComplexLinearSpace;
Lemma69:
the carrier of ((0). c1) = {(0. c1)}
by Def5;
reconsider c2 = the carrier of ((0). c1) --> 0 as Function of the carrier of ((0). c1), REAL by FUNCOP_1:57;
0. c1 is VECTOR of ((0). c1)
by Lemma69, TARSKI:def 1;
then Lemma70:
c2 . (0. c1) = 0
by FUNCOP_1:13;
Lemma71:
for b1 being VECTOR of ((0). c1)
for b2 being Complex holds c2 . (b2 * b1) = |.b2.| * (c2 . b1)
Lemma72:
for b1, b2 being VECTOR of ((0). c1) holds c2 . (b1 + b2) <= (c2 . b1) + (c2 . b2)
reconsider c3 = CNORMSTR(# the carrier of ((0). c1),the Zero of ((0). c1),the add of ((0). c1),the Mult of ((0). c1),c2 #) as non empty CNORMSTR by STRUCT_0:def 1;
:: deftheorem Def10 defines ComplexNormSpace-like CLVECT_1:def 10 :
theorem Th103: :: CLVECT_1:103
theorem Th104: :: CLVECT_1:104
theorem Th105: :: CLVECT_1:105
theorem Th106: :: CLVECT_1:106
theorem Th107: :: CLVECT_1:107
theorem Th108: :: CLVECT_1:108
theorem Th109: :: CLVECT_1:109
theorem Th110: :: CLVECT_1:110
theorem Th111: :: CLVECT_1:111
theorem Th112: :: CLVECT_1:112
theorem Th113: :: CLVECT_1:113
:: deftheorem Def11 defines + CLVECT_1:def 11 :
:: deftheorem Def12 defines - CLVECT_1:def 12 :
:: deftheorem Def13 defines - CLVECT_1:def 13 :
:: deftheorem Def14 defines * CLVECT_1:def 14 :
:: deftheorem Def15 defines convergent CLVECT_1:def 15 :
theorem Th114: :: CLVECT_1:114
canceled;
theorem Th115: :: CLVECT_1:115
theorem Th116: :: CLVECT_1:116
theorem Th117: :: CLVECT_1:117
theorem Th118: :: CLVECT_1:118
:: deftheorem Def16 defines ||. CLVECT_1:def 16 :
theorem Th119: :: CLVECT_1:119
:: deftheorem Def17 defines lim CLVECT_1:def 17 :
theorem Th120: :: CLVECT_1:120
theorem Th121: :: CLVECT_1:121
theorem Th122: :: CLVECT_1:122
theorem Th123: :: CLVECT_1:123
theorem Th124: :: CLVECT_1:124