:: HERMITAN semantic presentation
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Lm1:
0 = [**0,0**]
;
theorem Th1: :: HERMITAN:1
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theorem Th2: :: HERMITAN:2
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theorem :: HERMITAN:3
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theorem Th4: :: HERMITAN:4
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theorem Th5: :: HERMITAN:5
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i_FC *' =
(Re i_FC ) - ((Im i_FC ) * <i> )
by COMPLEX1:def 15
.=
(Re <i> ) - ((Im <i> ) * <i> )
.=
0 - (1 * <i> )
by COMPLEX1:17
;
then Lm2:
i_FC *' = <i> "
;
theorem :: HERMITAN:6
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canceled;
theorem Th7: :: HERMITAN:7
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theorem Th8: :: HERMITAN:8
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Lm3: the unity of F_Complex =
1r
by COMPLFLD:def 1
.=
1. F_Complex
by COMPLFLD:10
;
theorem Th9: :: HERMITAN:9
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theorem Th10: :: HERMITAN:10
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theorem Th11: :: HERMITAN:11
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theorem :: HERMITAN:12
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Lm4:
for a being Element of F_Complex holds Re a = Re (a *' )
by COMPLEX1:112;
theorem :: HERMITAN:13
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canceled;
theorem Th14: :: HERMITAN:14
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theorem Th15: :: HERMITAN:15
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theorem Th16: :: HERMITAN:16
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theorem Th17: :: HERMITAN:17
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theorem Th18: :: HERMITAN:18
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:: deftheorem Def1 defines cmplxhomogeneous HERMITAN:def 1 :
:: deftheorem Def2 defines *' HERMITAN:def 2 :
theorem :: HERMITAN:19
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theorem :: HERMITAN:20
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theorem Th21: :: HERMITAN:21
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theorem Th22: :: HERMITAN:22
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theorem :: HERMITAN:23
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theorem :: HERMITAN:24
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theorem Th25: :: HERMITAN:25
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theorem Th26: :: HERMITAN:26
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theorem :: HERMITAN:27
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theorem Th28: :: HERMITAN:28
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:: deftheorem defines QcFunctional HERMITAN:def 3 :
theorem Th29: :: HERMITAN:29
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:: deftheorem Def4 defines cmplxhomogeneousFAF HERMITAN:def 4 :
theorem Th30: :: HERMITAN:30
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:: deftheorem Def5 defines hermitan HERMITAN:def 5 :
:: deftheorem Def6 defines diagRvalued HERMITAN:def 6 :
:: deftheorem Def7 defines diagReR+0valued HERMITAN:def 7 :
Lm5:
now
let V be non
empty VectSpStr of
F_Complex ;
:: thesis: for f being Functional of V
for v1, w being Vector of V holds (FormFunctional f,(0Functional V)) . [v1,w] = 0. F_Complex let f be
Functional of
V;
:: thesis: for v1, w being Vector of V holds (FormFunctional f,(0Functional V)) . [v1,w] = 0. F_Complex set 0F =
0Functional V;
let v1,
w be
Vector of
V;
:: thesis: (FormFunctional f,(0Functional V)) . [v1,w] = 0. F_Complex thus (FormFunctional f,(0Functional V)) . [v1,w] =
(f . v1) * ((0Functional V) . w)
by BILINEAR:def 11
.=
(f . v1) * (0. F_Complex )
by HAHNBAN1:22
.=
0. F_Complex
by VECTSP_1:39
;
:: thesis: verum
end;
Lm6:
for V being non empty VectSpStr of F_Complex
for f being Functional of V holds FormFunctional f,(0Functional V) is hermitan
definition
let V,
W be non
empty VectSpStr of
F_Complex ;
let f be
Form of
V,
W;
func f *' -> Form of
V,
W means :
Def8:
:: HERMITAN:def 8
for
v being
Vector of
V for
w being
Vector of
W holds
it . [v,w] = (f . [v,w]) *' ;
existence
ex b1 being Form of V,W st
for v being Vector of V
for w being Vector of W holds b1 . [v,w] = (f . [v,w]) *'
uniqueness
for b1, b2 being Form of V,W st ( for v being Vector of V
for w being Vector of W holds b1 . [v,w] = (f . [v,w]) *' ) & ( for v being Vector of V
for w being Vector of W holds b2 . [v,w] = (f . [v,w]) *' ) holds
b1 = b2
end;
:: deftheorem Def8 defines *' HERMITAN:def 8 :
theorem Th31: :: HERMITAN:31
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theorem :: HERMITAN:32
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theorem :: HERMITAN:33
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theorem Th34: :: HERMITAN:34
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theorem Th35: :: HERMITAN:35
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theorem :: HERMITAN:36
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theorem :: HERMITAN:37
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theorem Th38: :: HERMITAN:38
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theorem Th39: :: HERMITAN:39
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theorem Th40: :: HERMITAN:40
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for
V,
W being non
empty add-associative right_zeroed right_complementable VectSp-like VectSpStr of
F_Complex for
v,
u being
Vector of
V for
w,
t being
Vector of
W for
a,
b being
Element of
F_Complex for
f being
sesquilinear-Form of
V,
W holds
f . [(v + (a * u)),(w + (b * t))] = ((f . [v,w]) + ((b *' ) * (f . [v,t]))) + ((a * (f . [u,w])) + (a * ((b *' ) * (f . [u,t]))))
theorem Th41: :: HERMITAN:41
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for
V,
W being
VectSp of
F_Complex for
v,
u being
Vector of
V for
w,
t being
Vector of
W for
a,
b being
Element of
F_Complex for
f being
sesquilinear-Form of
V,
W holds
f . [(v - (a * u)),(w - (b * t))] = ((f . [v,w]) - ((b *' ) * (f . [v,t]))) - ((a * (f . [u,w])) - (a * ((b *' ) * (f . [u,t]))))
theorem Th42: :: HERMITAN:42
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theorem :: HERMITAN:43
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:: deftheorem defines signnorm HERMITAN:def 9 :
theorem Th44: :: HERMITAN:44
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theorem Th45: :: HERMITAN:45
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for
V being
VectSp of
F_Complex for
v,
w being
Vector of
V for
f being
sesquilinear-Form of
V,
V for
r being
Real for
a being
Element of
F_Complex st
|.a.| = 1 &
Re (a * (f . [w,v])) = |.(f . [w,v]).| &
Im (a * (f . [w,v])) = 0 holds
f . [(v - (([**r,0**] * a) * w)),(v - (([**r,0**] * a) * w))] = (((f . [v,v]) - ([**r,0**] * (a * (f . [w,v])))) - ([**r,0**] * ((a *' ) * (f . [v,w])))) + ([**(r ^2 ),0**] * (f . [w,w]))
theorem Th46: :: HERMITAN:46
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for
V being
VectSp of
F_Complex for
v,
w being
Vector of
V for
f being
diagReR+0valued hermitan-Form of
V for
r being
Real for
a being
Element of
F_Complex st
|.a.| = 1 &
Re (a * (f . [w,v])) = |.(f . [w,v]).| &
Im (a * (f . [w,v])) = 0 holds
(
Re (f . [(v - (([**r,0**] * a) * w)),(v - (([**r,0**] * a) * w))]) = ((signnorm f,v) - ((2 * |.(f . [w,v]).|) * r)) + ((signnorm f,w) * (r ^2 )) & 0
<= ((signnorm f,v) - ((2 * |.(f . [w,v]).|) * r)) + ((signnorm f,w) * (r ^2 )) )
theorem Th47: :: HERMITAN:47
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theorem Th48: :: HERMITAN:48
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theorem Th49: :: HERMITAN:49
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theorem Th50: :: HERMITAN:50
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theorem :: HERMITAN:51
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theorem Th52: :: HERMITAN:52
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theorem :: HERMITAN:53
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:: deftheorem Def10 defines quasinorm HERMITAN:def 10 :
theorem :: HERMITAN:54
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theorem Th55: :: HERMITAN:55
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theorem Th56: :: HERMITAN:56
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theorem :: HERMITAN:57
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theorem Th58: :: HERMITAN:58
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theorem Th59: :: HERMITAN:59
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theorem Th60: :: HERMITAN:60
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theorem Th61: :: HERMITAN:61
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:: deftheorem defines RQ*Form HERMITAN:def 11 :
theorem Th62: :: HERMITAN:62
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definition
let V,
W be
VectSp of
F_Complex ;
let f be
sesquilinear-Form of
V,
W;
func Q*Form f -> sesquilinear-Form of
(VectQuot V,(LKer f)),
(VectQuot W,(RKer (f *' ))) means :
Def12:
:: HERMITAN:def 12
for
A being
Vector of
(VectQuot V,(LKer f)) for
B being
Vector of
(VectQuot W,(RKer (f *' ))) for
v being
Vector of
V for
w being
Vector of
W st
A = v + (LKer f) &
B = w + (RKer (f *' )) holds
it . [A,B] = f . [v,w];
existence
ex b1 being sesquilinear-Form of (VectQuot V,(LKer f)),(VectQuot W,(RKer (f *' ))) st
for A being Vector of (VectQuot V,(LKer f))
for B being Vector of (VectQuot W,(RKer (f *' )))
for v being Vector of V
for w being Vector of W st A = v + (LKer f) & B = w + (RKer (f *' )) holds
b1 . [A,B] = f . [v,w]
uniqueness
for b1, b2 being sesquilinear-Form of (VectQuot V,(LKer f)),(VectQuot W,(RKer (f *' ))) st ( for A being Vector of (VectQuot V,(LKer f))
for B being Vector of (VectQuot W,(RKer (f *' )))
for v being Vector of V
for w being Vector of W st A = v + (LKer f) & B = w + (RKer (f *' )) holds
b1 . [A,B] = f . [v,w] ) & ( for A being Vector of (VectQuot V,(LKer f))
for B being Vector of (VectQuot W,(RKer (f *' )))
for v being Vector of V
for w being Vector of W st A = v + (LKer f) & B = w + (RKer (f *' )) holds
b2 . [A,B] = f . [v,w] ) holds
b1 = b2
end;
:: deftheorem Def12 defines Q*Form HERMITAN:def 12 :
theorem Th63: :: HERMITAN:63
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theorem Th64: :: HERMITAN:64
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theorem Th65: :: HERMITAN:65
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theorem Th66: :: HERMITAN:66
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theorem Th67: :: HERMITAN:67
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:: deftheorem Def13 defines positivediagvalued HERMITAN:def 13 :
:: deftheorem defines ScalarForm HERMITAN:def 14 :
theorem :: HERMITAN:68
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theorem Th69: :: HERMITAN:69
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theorem Th70: :: HERMITAN:70
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