:: CLVECT_1 semantic presentation
:: deftheorem defines * CLVECT_1:def 1 :
Lm1:
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 A1:
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;
A2:
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
thus
for
x,
y,
z being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
(x + y) + z = x + (y + z)
proof
let x,
y,
z be
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #);
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 A1;
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 #)) = H1(
X,
O)
by A1;
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 #)
thus
for
z being
Complex for
x,
y being
VECTOR of
CLSStruct(#
ZS,
O,
F,
G #) holds
z * (x + y) = (z * x) + (z * y)
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 :: 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 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 :: CLVECT_1:6
theorem Th7: :: CLVECT_1:7
theorem Th8: :: CLVECT_1:8
theorem :: CLVECT_1:9
theorem Th10: :: CLVECT_1:10
theorem Th11: :: CLVECT_1:11
theorem :: CLVECT_1:12
theorem :: CLVECT_1:13
Lm2:
for V being non empty LoopStr holds Sum (<*> the carrier of V) = 0. V
Lm3:
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 :: CLVECT_1:14
theorem :: CLVECT_1:15
theorem :: CLVECT_1:16
theorem :: CLVECT_1:17
Lm4:
1r + 1r = [*2,0*]
theorem Th18: :: CLVECT_1:18
theorem :: CLVECT_1:19
theorem :: CLVECT_1:20
:: deftheorem Def3 defines lineary-closed CLVECT_1:def 3 :
theorem Th21: :: CLVECT_1:21
theorem Th22: :: CLVECT_1:22
theorem :: CLVECT_1:23
theorem Th24: :: CLVECT_1:24
theorem :: CLVECT_1:25
theorem :: CLVECT_1:26
theorem :: CLVECT_1:27
:: deftheorem Def4 defines Subspace CLVECT_1:def 4 :
theorem :: CLVECT_1:28
theorem Th29: :: CLVECT_1:29
theorem Th30: :: CLVECT_1:30
theorem Th31: :: CLVECT_1:31
theorem :: CLVECT_1:32
theorem Th33: :: CLVECT_1:33
theorem Th34: :: CLVECT_1:34
theorem Th35: :: CLVECT_1:35
theorem Th36: :: CLVECT_1:36
Lm5:
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 :: CLVECT_1:38
theorem :: 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 :: CLVECT_1:49
theorem Th50: :: CLVECT_1:50
theorem Th51: :: CLVECT_1:51
theorem :: CLVECT_1:52
theorem :: CLVECT_1:53
theorem :: CLVECT_1:54
theorem Th55: :: CLVECT_1:55
:: deftheorem Def5 defines (0). CLVECT_1:def 5 :
:: deftheorem defines (Omega). CLVECT_1:def 6 :
theorem Th56: :: CLVECT_1:56
theorem Th57: :: CLVECT_1:57
theorem :: CLVECT_1:58
theorem :: CLVECT_1:59
theorem :: CLVECT_1:60
theorem :: CLVECT_1:61
:: deftheorem defines + CLVECT_1:def 7 :
Lm6:
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 :: CLVECT_1:64
theorem Th65: :: CLVECT_1:65
Lm7:
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 :: CLVECT_1:68
theorem Th69: :: CLVECT_1:69
theorem Th70: :: CLVECT_1:70
theorem Th71: :: CLVECT_1:71
theorem Th72: :: CLVECT_1:72
theorem :: CLVECT_1:73
theorem Th74: :: CLVECT_1:74
theorem Th75: :: CLVECT_1:75
theorem Th76: :: CLVECT_1:76
theorem :: CLVECT_1:77
theorem Th78: :: CLVECT_1:78
theorem Th79: :: CLVECT_1:79
theorem :: CLVECT_1:80
theorem Th81: :: CLVECT_1:81
theorem :: CLVECT_1:82
theorem Th83: :: CLVECT_1:83
theorem :: 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 :: CLVECT_1:90
theorem :: CLVECT_1:91
theorem :: CLVECT_1:92
theorem :: CLVECT_1:93
theorem :: CLVECT_1:94
theorem :: CLVECT_1:95
theorem :: CLVECT_1:96
theorem :: CLVECT_1:97
theorem Th98: :: CLVECT_1:98
theorem :: CLVECT_1:99
theorem :: CLVECT_1:100
theorem :: CLVECT_1:101
theorem :: CLVECT_1:102
deffunc H1( CNORMSTR ) -> Element of the carrier of $1 = 0. $1;
:: deftheorem defines ||. CLVECT_1:def 9 :
consider V being ComplexLinearSpace;
Lm8:
the carrier of ((0). V) = {(0. V)}
by Def5;
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 Lm8, TARSKI:def 1;
then Lm9:
niltonil . (0. V) = 0
by FUNCOP_1:13;
Lm10:
for u being VECTOR of ((0). V)
for z being Complex holds niltonil . (z * u) = |.z.| * (niltonil . u)
Lm11:
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 :: CLVECT_1:103
theorem Th104: :: CLVECT_1:104
theorem Th105: :: CLVECT_1:105
theorem Th106: :: CLVECT_1:106
theorem :: 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 :: 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 :: 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 :: CLVECT_1:120
theorem :: CLVECT_1:121
theorem :: CLVECT_1:122
theorem :: CLVECT_1:123
theorem :: CLVECT_1:124