:: HERMITAN semantic presentation
Lemma1:
0 = [**0,0**]
;
theorem Th1: :: HERMITAN:1
theorem Th2: :: HERMITAN:2
theorem Th3: :: HERMITAN:3
theorem Th4: :: HERMITAN:4
theorem Th5: :: HERMITAN:5
i_FC *' =
(Re i_FC ) - ((Im i_FC ) * <i> )
.=
(Re <i> ) - ((Im <i> ) * <i> )
.=
0 - (1 * <i> )
by COMPLEX1:17
;
then Lemma6:
i_FC *' = <i> "
;
theorem Th6: :: HERMITAN:6
canceled;
theorem Th7: :: HERMITAN:7
theorem Th8: :: HERMITAN:8
E9: the unity of F_Complex =
1r
by COMPLFLD:def 1
.=
1. F_Complex
by COMPLFLD:10
;
theorem Th9: :: HERMITAN:9
theorem Th10: :: HERMITAN:10
theorem Th11: :: HERMITAN:11
theorem Th12: :: HERMITAN:12
Lemma13:
for b1 being Element of F_Complex holds Re b1 = Re (b1 *' )
by COMPLEX1:112;
theorem Th13: :: HERMITAN:13
canceled;
theorem Th14: :: HERMITAN:14
theorem Th15: :: HERMITAN:15
theorem Th16: :: HERMITAN:16
theorem Th17: :: HERMITAN:17
theorem Th18: :: HERMITAN:18
:: deftheorem Def1 defines cmplxhomogeneous HERMITAN:def 1 :
:: deftheorem Def2 defines *' HERMITAN:def 2 :
theorem Th19: :: HERMITAN:19
theorem Th20: :: HERMITAN:20
theorem Th21: :: HERMITAN:21
theorem Th22: :: HERMITAN:22
theorem Th23: :: HERMITAN:23
theorem Th24: :: HERMITAN:24
theorem Th25: :: HERMITAN:25
theorem Th26: :: HERMITAN:26
theorem Th27: :: HERMITAN:27
theorem Th28: :: HERMITAN:28
:: deftheorem Def3 defines QcFunctional HERMITAN:def 3 :
theorem Th29: :: HERMITAN:29
:: deftheorem Def4 defines cmplxhomogeneousFAF HERMITAN:def 4 :
theorem Th30: :: HERMITAN:30
:: deftheorem Def5 defines hermitan HERMITAN:def 5 :
:: deftheorem Def6 defines diagRvalued HERMITAN:def 6 :
:: deftheorem Def7 defines diagReR+0valued HERMITAN:def 7 :
Lemma33:
for b1 being non empty VectSpStr of F_Complex
for b2 being Functional of b1 holds FormFunctional b2,(0Functional b1) is hermitan
definition
let c
1, c
2 be non
empty VectSpStr of
F_Complex ;
let c
3 be
Form of c
1,c
2;
func c
3 *' -> Form of a
1,a
2 means :
Def8:
:: HERMITAN:def 8
for b
1 being
Vector of a
1for b
2 being
Vector of a
2 holds a
4 . b
1,b
2 = (a3 . b1,b2) *' ;
existence
ex b1 being Form of c1,c2 st
for b2 being Vector of c1
for b3 being Vector of c2 holds b1 . b2,b3 = (c3 . b2,b3) *'
uniqueness
for b1, b2 being Form of c1,c2 holds
( ( for b3 being Vector of c1
for b4 being Vector of c2 holds b1 . b3,b4 = (c3 . b3,b4) *' ) & ( for b3 being Vector of c1
for b4 being Vector of c2 holds b2 . b3,b4 = (c3 . b3,b4) *' ) implies b1 = b2 )
end;
:: deftheorem Def8 defines *' HERMITAN:def 8 :
theorem Th31: :: HERMITAN:31
theorem Th32: :: HERMITAN:32
theorem Th33: :: HERMITAN:33
theorem Th34: :: HERMITAN:34
theorem Th35: :: HERMITAN:35
theorem Th36: :: HERMITAN:36
theorem Th37: :: HERMITAN:37
theorem Th38: :: HERMITAN:38
theorem Th39: :: HERMITAN:39
theorem Th40: :: HERMITAN:40
for b
1, b
2 being non
empty add-associative right_zeroed right_complementable VectSp-like VectSpStr of
F_Complex for b
3, b
4 being
Vector of b
1for b
5, b
6 being
Vector of b
2for b
7, b
8 being
Element of
F_Complex for b
9 being
sesquilinear-Form of b
1,b
2 holds b
9 . (b3 + (b7 * b4)),
(b5 + (b8 * b6)) = ((b9 . b3,b5) + ((b8 *' ) * (b9 . b3,b6))) + ((b7 * (b9 . b4,b5)) + (b7 * ((b8 *' ) * (b9 . b4,b6))))
theorem Th41: :: HERMITAN:41
for b
1, b
2 being
VectSp of
F_Complex for b
3, b
4 being
Vector of b
1for b
5, b
6 being
Vector of b
2for b
7, b
8 being
Element of
F_Complex for b
9 being
sesquilinear-Form of b
1,b
2 holds b
9 . (b3 - (b7 * b4)),
(b5 - (b8 * b6)) = ((b9 . b3,b5) - ((b8 *' ) * (b9 . b3,b6))) - ((b7 * (b9 . b4,b5)) - (b7 * ((b8 *' ) * (b9 . b4,b6))))
theorem Th42: :: HERMITAN:42
theorem Th43: :: HERMITAN:43
:: deftheorem Def9 defines signnorm HERMITAN:def 9 :
theorem Th44: :: HERMITAN:44
theorem Th45: :: HERMITAN:45
for b
1 being
VectSp of
F_Complex for b
2, b
3 being
Vector of b
1for b
4 being
sesquilinear-Form of b
1,b
1for b
5 being
Realfor b
6 being
Element of
F_Complex holds
(
|.b6.| = 1 &
Re (b6 * (b4 . b3,b2)) = |.(b4 . b3,b2).| &
Im (b6 * (b4 . b3,b2)) = 0 implies b
4 . (b2 - (([**b5,0**] * b6) * b3)),
(b2 - (([**b5,0**] * b6) * b3)) = (((b4 . b2,b2) - ([**b5,0**] * (b6 * (b4 . b3,b2)))) - ([**b5,0**] * ((b6 *' ) * (b4 . b2,b3)))) + ([**(b5 ^2 ),0**] * (b4 . b3,b3)) )
theorem Th46: :: HERMITAN:46
for b
1 being
VectSp of
F_Complex for b
2, b
3 being
Vector of b
1for b
4 being
diagReR+0valued hermitan-Form of b
1for b
5 being
Realfor b
6 being
Element of
F_Complex holds
(
|.b6.| = 1 &
Re (b6 * (b4 . b3,b2)) = |.(b4 . b3,b2).| &
Im (b6 * (b4 . b3,b2)) = 0 implies (
Re (b4 . (b2 - (([**b5,0**] * b6) * b3)),(b2 - (([**b5,0**] * b6) * b3))) = ((signnorm b4,b2) - ((2 * |.(b4 . b3,b2).|) * b5)) + ((signnorm b4,b3) * (b5 ^2 )) & 0
<= ((signnorm b4,b2) - ((2 * |.(b4 . b3,b2).|) * b5)) + ((signnorm b4,b3) * (b5 ^2 )) ) )
theorem Th47: :: HERMITAN:47
theorem Th48: :: HERMITAN:48
theorem Th49: :: HERMITAN:49
theorem Th50: :: HERMITAN:50
theorem Th51: :: HERMITAN:51
theorem Th52: :: HERMITAN:52
theorem Th53: :: HERMITAN:53
:: deftheorem Def10 defines quasinorm HERMITAN:def 10 :
theorem Th54: :: HERMITAN:54
theorem Th55: :: HERMITAN:55
theorem Th56: :: HERMITAN:56
theorem Th57: :: HERMITAN:57
theorem Th58: :: HERMITAN:58
theorem Th59: :: HERMITAN:59
theorem Th60: :: HERMITAN:60
theorem Th61: :: HERMITAN:61
:: deftheorem Def11 defines RQ*Form HERMITAN:def 11 :
theorem Th62: :: HERMITAN:62
definition
let c
1, c
2 be
VectSp of
F_Complex ;
let c
3 be
sesquilinear-Form of c
1,c
2;
func Q*Form c
3 -> sesquilinear-Form of
(VectQuot a1,(LKer a3)),
(VectQuot a2,(RKer (a3 *' ))) means :
Def12:
:: HERMITAN:def 12
for b
1 being
Vector of
(VectQuot a1,(LKer a3))for b
2 being
Vector of
(VectQuot a2,(RKer (a3 *' )))for b
3 being
Vector of a
1for b
4 being
Vector of a
2 holds
( b
1 = b
3 + (LKer a3) & b
2 = b
4 + (RKer (a3 *' )) implies a
4 . b
1,b
2 = a
3 . b
3,b
4 );
existence
ex b1 being sesquilinear-Form of (VectQuot c1,(LKer c3)),(VectQuot c2,(RKer (c3 *' ))) st
for b2 being Vector of (VectQuot c1,(LKer c3))
for b3 being Vector of (VectQuot c2,(RKer (c3 *' )))
for b4 being Vector of c1
for b5 being Vector of c2 holds
( b2 = b4 + (LKer c3) & b3 = b5 + (RKer (c3 *' )) implies b1 . b2,b3 = c3 . b4,b5 )
uniqueness
for b1, b2 being sesquilinear-Form of (VectQuot c1,(LKer c3)),(VectQuot c2,(RKer (c3 *' ))) holds
( ( for b3 being Vector of (VectQuot c1,(LKer c3))
for b4 being Vector of (VectQuot c2,(RKer (c3 *' )))
for b5 being Vector of c1
for b6 being Vector of c2 holds
( b3 = b5 + (LKer c3) & b4 = b6 + (RKer (c3 *' )) implies b1 . b3,b4 = c3 . b5,b6 ) ) & ( for b3 being Vector of (VectQuot c1,(LKer c3))
for b4 being Vector of (VectQuot c2,(RKer (c3 *' )))
for b5 being Vector of c1
for b6 being Vector of c2 holds
( b3 = b5 + (LKer c3) & b4 = b6 + (RKer (c3 *' )) implies b2 . b3,b4 = c3 . b5,b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines Q*Form HERMITAN:def 12 :
theorem Th63: :: HERMITAN:63
theorem Th64: :: HERMITAN:64
theorem Th65: :: HERMITAN:65
theorem Th66: :: HERMITAN:66
theorem Th67: :: HERMITAN:67
:: deftheorem Def13 defines positivediagvalued HERMITAN:def 13 :
:: deftheorem Def14 defines ScalarForm HERMITAN:def 14 :
theorem Th68: :: HERMITAN:68
theorem Th69: :: HERMITAN:69
theorem Th70: :: HERMITAN:70