:: CLVECT_1 semantic presentation
:: deftheorem Def1 defines * CLVECT_1:def 1 :
E26:
now
take ZS =
{0};
reconsider O = 0 as
Element of
ZS by TARSKI:def 1;
take O =
O;
deffunc H1(
Element of
ZS,
Element of
ZS)
-> Element of
ZS =
O;
consider F being
BinOp of
ZS such that E28:
for
x,
y being
Element of
ZS holds
F . x,
y = H1(
x,
y)
from BINOP_1:sch 4();
reconsider G =
[:COMPLEX ,ZS:] --> O as
Function of
[:COMPLEX ,ZS:],
ZS by FUNCOP_1:57;
E31:
for
a being
Element of
COMPLEX for
x being
Element of
ZS holds
G . [a,x] = O
take F =
F;
take G =
G;
set W =
CLSStruct(#
ZS,
O,
F,
G #);
thus
for
x,
y being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
x + y = y + x
proof
let x be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #),
y be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #);
E34:
(
x + y = F . x,
y &
y + x = F . y,
x )
;
reconsider X =
x,
Y =
y as
Element of
ZS ;
(
x + y = H1(
X,
Y) &
y + x = H1(
Y,
X) )
by , ;
hence
x + y = y + x
;
end;
thus
for
x,
y,
z being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
(x + y) + z = x + (y + z)
proof
let x be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #),
y be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #),
z be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #);
E37:
(
(x + y) + z = F . (x + y),
z &
x + (y + z) = F . x,
(y + z) )
;
reconsider X =
x,
Y =
y,
Z =
z as
Element of
ZS ;
(
(x + y) + z = H1(
H1(
X,
Y),
Z) &
x + (y + z) = H1(
X,
H1(
Y,
Z)) )
by , ;
hence
(x + y) + z = x + (y + z)
;
end;
thus
for
x being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
x + (0. CLSStruct(# ZS,O,F,G #)) = x
proof
let x be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #);
reconsider X =
x as
Element of
ZS ;
x + (0. CLSStruct(# ZS,O,F,G #)) =
F . x,
(0. CLSStruct(# ZS,O,F,G #))
.=
H1(
X,
O)
by
;
hence
x + (0. CLSStruct(# ZS,O,F,G #)) = x
by TARSKI:def 1;
end;
thus
for
x being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) ex
y being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) st
x + y = 0. CLSStruct(#
ZS,
O,
F,
G #)
proof
let x be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #);
reconsider y =
O as
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) ;
take
y
;
thus x + y =
F . x,
y
.=
0. CLSStruct(#
ZS,
O,
F,
G #)
by
;
end;
thus
for
z being
Complex for
x,
y being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
z * (x + y) = (z * x) + (z * y)
proof
let z be
Complex;
let x be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #),
y be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #);
reconsider X =
x,
Y =
y as
Element of
ZS ;
(z * x) + (z * y) =
F . (z * x),
(z * y)
.=
H1(
O,
O)
by
;
hence
z * (x + y) = (z * x) + (z * y)
by ;
end;
thus
for
z1,
z2 being
Complex for
x being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
(z1 + z2) * x = (z1 * x) + (z2 * x)
thus
for
z1,
z2 being
Complex for
x being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
(z1 * z2) * x = z1 * (z2 * x)
thus
for
x being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
1r * x = x
end;
:: deftheorem Def2 defines ComplexLinearSpace-like CLVECT_1:def 2 :
theorem Th1: :: CLVECT_1:1
for
V being non
empty CLSStruct st ( for
v,
w being
VECTOR of
V holds
v + w = w + v ) & ( for
u,
v,
w being
VECTOR of
V holds
(u + v) + w = u + (v + w) ) & ( for
v being
VECTOR of
V holds
v + (0. V) = v ) & ( for
v being
VECTOR of
V ex
w being
VECTOR of
V st
v + w = 0. V ) & ( for
z being
Complex for
v,
w being
VECTOR of
V holds
z * (v + w) = (z * v) + (z * w) ) & ( for
z1,
z2 being
Complex for
v being
VECTOR of
V holds
(z1 + z2) * v = (z1 * v) + (z2 * v) ) & ( for
z1,
z2 being
Complex for
v being
VECTOR of
V holds
(z1 * z2) * v = z1 * (z2 * v) ) & ( for
v being
VECTOR of
V holds
1r * v = v ) holds
V is
ComplexLinearSpace by , 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
Lemma61:
for V being non empty LoopStr holds Sum (<*> the carrier of V) = 0. V
Lemma65:
for V being non empty LoopStr
for F being FinSequence of the carrier of V st len F = 0 holds
Sum F = 0. V
theorem Th14: :: CLVECT_1:14
theorem Th15: :: CLVECT_1:15
theorem Th16: :: CLVECT_1:16
theorem Th17: :: CLVECT_1:17
Lemma85:
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
Lemma110:
for V being ComplexLinearSpace
for V1 being Subset of V
for W being Subspace of V st the carrier of W = V1 holds
V1 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 :
Lemma156:
for V being ComplexLinearSpace
for W being Subspace of V holds (0. V) + W = the carrier of W
:: 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
Lemma163:
for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( v in W iff v + W = the carrier of W )
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 V being ComplexLinearSpace;
Lemma186:
the carrier of ((0). V) = {(0. V)}
by ;
reconsider niltonil = the carrier of ((0). V) --> 0 as Function of the carrier of ((0). V), REAL by FUNCOP_1:57;
0. V is VECTOR of ((0). V)
by , TARSKI:def 1;
then Lemma188:
niltonil . (0. V) = 0
by FUNCOP_1:13;
Lemma189:
for u being VECTOR of ((0). V)
for z being Complex holds niltonil . (z * u) = |.z.| * (niltonil . u)
Lemma190:
for u, v being VECTOR of ((0). V) holds niltonil . (u + v) <= (niltonil . u) + (niltonil . v)
reconsider X = CNORMSTR(# the carrier of ((0). V),the Zero of ((0). V),the add of ((0). V),the Mult of ((0). V),niltonil #) 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