:: Complex Linear Space and Complex Normed Space
:: by Noboru Endou
::
:: Received January 26, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

definition

attr c₁ is strict ;
struct CLSStruct -> addLoopStr ;
aggr CLSStruct(# carrier, ZeroF, addF, Mult #) -> CLSStruct ;
sel Mult c₁ -> Function of [:COMPLEX, the carrier of c₁:], the carrier of c₁;

end;

registration

cluster non empty for CLSStruct ;

existence
not for b₁ being CLSStruct holds b₁ is empty

proof end;

end;

definition

let V be CLSStruct ;
mode VECTOR of V is Element of V;

end;

definition

let V be non empty CLSStruct ;
let v be VECTOR of V;
let z be Complex;

func z * v -> Element of V equals :: CLVECT_1:def 1
the Mult of V . [z,v];

coherence
the Mult of V . [z,v] is Element of V

proof end;

end;

:: deftheorem defines * CLVECT_1:def 1 :
for V being non empty CLSStruct
for v being VECTOR of V
for z being Complex holds z * v = the Mult of V . [z,v];

registration

let ZS be non empty set ;
let O be Element of ZS;
let F be BinOp of ZS;
let G be Function of [:COMPLEX,ZS:],ZS;

cluster CLSStruct(# ZS,O,F,G #) -> non empty ;

coherence
not CLSStruct(# ZS,O,F,G #) is empty ;

end;

definition

let IT be non empty CLSStruct ;

attr IT is vector-distributive means :Def2: :: CLVECT_1:def 2
for a being Complex
for v, w being VECTOR of IT holds a * (v + w) = (a * v) + (a * w);

attr IT is scalar-distributive means :Def3: :: CLVECT_1:def 3
for a, b being Complex
for v being VECTOR of IT holds (a + b) * v = (a * v) + (b * v);

attr IT is scalar-associative means :Def4: :: CLVECT_1:def 4
for a, b being Complex
for v being VECTOR of IT holds (a * b) * v = a * (b * v);

attr IT is scalar-unital means :Def5: :: CLVECT_1:def 5
for v being VECTOR of IT holds 1r * v = v;

end;

:: deftheorem Def2 defines vector-distributive CLVECT_1:def 2 :
for IT being non empty CLSStruct holds
( IT is vector-distributive iff for a being Complex
for v, w being VECTOR of IT holds a * (v + w) = (a * v) + (a * w) );

:: deftheorem Def3 defines scalar-distributive CLVECT_1:def 3 :
for IT being non empty CLSStruct holds
( IT is scalar-distributive iff for a, b being Complex
for v being VECTOR of IT holds (a + b) * v = (a * v) + (b * v) );

:: deftheorem Def4 defines scalar-associative CLVECT_1:def 4 :
for IT being non empty CLSStruct holds
( IT is scalar-associative iff for a, b being Complex
for v being VECTOR of IT holds (a * b) * v = a * (b * v) );

:: deftheorem Def5 defines scalar-unital CLVECT_1:def 5 :
for IT being non empty CLSStruct holds
( IT is scalar-unital iff for v being VECTOR of IT holds 1r * v = v );

definition

func Trivial-CLSStruct -> strict CLSStruct equals :: CLVECT_1:def 6
CLSStruct(# {0},op0,op2,(pr2 (COMPLEX,{0})) #);

coherence
CLSStruct(# {0},op0,op2,(pr2 (COMPLEX,{0})) #) is strict CLSStruct ;

end;

:: deftheorem defines Trivial-CLSStruct CLVECT_1:def 6 :
Trivial-CLSStruct = CLSStruct(# {0},op0,op2,(pr2 (COMPLEX,{0})) #);

registration

cluster Trivial-CLSStruct -> 1 -element strict ;

coherence
Trivial-CLSStruct is 1 -element ;

end;

registration

cluster non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for CLSStruct ;

existence
ex b₁ being non empty CLSStruct st
( b₁ is strict & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is vector-distributive & b₁ is scalar-distributive & b₁ is scalar-associative & b₁ is scalar-unital )

proof end;

end;

definition

mode ComplexLinearSpace is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct ;

end;

theorem Th1: :: CLVECT_1:1

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex st ( z = 0 or v = 0. V ) holds
z * v = 0. V

proof end;

theorem Th2: :: CLVECT_1:2

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex holds
( not z * v = 0. V or z = 0 or v = 0. V )

proof end;

theorem Th3: :: CLVECT_1:3

for V being ComplexLinearSpace
for v being VECTOR of V holds - v = (- 1r) * v

proof end;

theorem Th4: :: CLVECT_1:4

for V being ComplexLinearSpace
for v being VECTOR of V st v = - v holds
v = 0. V

proof end;

theorem :: CLVECT_1:5

for V being ComplexLinearSpace
for v being VECTOR of V st v + v = 0. V holds
v = 0. V

proof end;

theorem Th6: :: CLVECT_1:6

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex holds z * (- v) = (- z) * v

proof end;

theorem Th7: :: CLVECT_1:7

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex holds z * (- v) = - (z * v)

proof end;

theorem :: CLVECT_1:8

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex holds (- z) * (- v) = z * v

proof end;

theorem Th9: :: CLVECT_1:9

for V being ComplexLinearSpace
for u, v being VECTOR of V
for z being Complex holds z * (v - u) = (z * v) - (z * u)

proof end;

theorem Th10: :: CLVECT_1:10

for V being ComplexLinearSpace
for v being VECTOR of V
for z1, z2 being Complex holds (z1 - z2) * v = (z1 * v) - (z2 * v)

proof end;

theorem :: CLVECT_1:11

for V being ComplexLinearSpace
for u, v being VECTOR of V
for z being Complex st z <> 0 & z * v = z * u holds
v = u

proof end;

theorem :: CLVECT_1:12

for V being ComplexLinearSpace
for v being VECTOR of V
for z1, z2 being Complex st v <> 0. V & z1 * v = z2 * v holds
z1 = z2

proof end;

Lm1: for V being non empty addLoopStr holds Sum (<*> the carrier of V) = 0. V

proof end;

Lm2: for V being non empty addLoopStr
for F being FinSequence of the carrier of V st len F = 0 holds
Sum F = 0. V

proof end;

theorem :: CLVECT_1:13

for V being ComplexLinearSpace
for z being Complex
for F, G being FinSequence of the carrier of V st len F = len G & ( for k being Nat
for v being VECTOR of V st k in dom F & v = G . k holds
F . k = z * v ) holds
Sum F = z * (Sum G)

proof end;

theorem :: CLVECT_1:14

for V being ComplexLinearSpace
for z being Complex holds z * (Sum (<*> the carrier of V)) = 0. V by Lm1, Th1;

theorem :: CLVECT_1:15

for V being ComplexLinearSpace
for u, v being VECTOR of V
for z being Complex holds z * (Sum <*v,u*>) = (z * v) + (z * u)

proof end;

theorem :: CLVECT_1:16

for V being ComplexLinearSpace
for u, v1, v2 being VECTOR of V
for z being Complex holds z * (Sum <*u,v1,v2*>) = ((z * u) + (z * v1)) + (z * v2)

proof end;

theorem Th17: :: CLVECT_1:17

for V being ComplexLinearSpace
for v being VECTOR of V holds Sum <*v,v*> = (1r + 1r) * v

proof end;

theorem :: CLVECT_1:18

for V being ComplexLinearSpace
for v being VECTOR of V holds Sum <*(- v),(- v)*> = (- (1r + 1r)) * v

proof end;

theorem :: CLVECT_1:19

for V being ComplexLinearSpace
for v being VECTOR of V holds Sum <*v,v,v*> = ((1r + 1r) + 1r) * v

proof end;

definition

let V be ComplexLinearSpace;
let V1 be Subset of V;

attr V1 is linearly-closed means :: CLVECT_1:def 7
( ( for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for z being Complex
for v being VECTOR of V st v in V1 holds
z * v in V1 ) );

end;

:: deftheorem defines linearly-closed CLVECT_1:def 7 :
for V being ComplexLinearSpace
for V1 being Subset of V holds
( V1 is linearly-closed iff ( ( for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for z being Complex
for v being VECTOR of V st v in V1 holds
z * v in V1 ) ) );

theorem Th20: :: CLVECT_1:20

for V being ComplexLinearSpace
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
0. V in V1

proof end;

theorem Th21: :: CLVECT_1:21

for V being ComplexLinearSpace
for V1 being Subset of V st V1 is linearly-closed holds
for v being VECTOR of V st v in V1 holds
- v in V1

proof end;

theorem :: CLVECT_1:22

for V being ComplexLinearSpace
for V1 being Subset of V st V1 is linearly-closed holds
for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1

proof end;

theorem Th23: :: CLVECT_1:23

for V being ComplexLinearSpace holds {(0. V)} is linearly-closed

proof end;

theorem :: CLVECT_1:24

for V being ComplexLinearSpace
for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed ;

theorem :: CLVECT_1:25

for V being ComplexLinearSpace
for V1, V2, V3 being Subset of V st V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where u, v is VECTOR of V : ( v in V1 & u in V2 ) } holds
V3 is linearly-closed

proof end;

theorem :: CLVECT_1:26

for V being ComplexLinearSpace
for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed

proof end;

definition

let V be ComplexLinearSpace;

mode Subspace of V -> ComplexLinearSpace means :Def8: :: CLVECT_1:def 8
( the carrier of it c= the carrier of V & 0. it = 0. V & the addF of it = the addF of V || the carrier of it & the Mult of it = the Mult of V | [:COMPLEX, the carrier of it:] );

existence
ex b₁ being ComplexLinearSpace st
( the carrier of b₁ c= the carrier of V & 0. b₁ = 0. V & the addF of b₁ = the addF of V || the carrier of b₁ & the Mult of b₁ = the Mult of V | [:COMPLEX, the carrier of b₁:] )

proof end;

end;

:: deftheorem Def8 defines Subspace CLVECT_1:def 8 :
for V, b₂ being ComplexLinearSpace holds
( b₂ is Subspace of V iff ( the carrier of b₂ c= the carrier of V & 0. b₂ = 0. V & the addF of b₂ = the addF of V || the carrier of b₂ & the Mult of b₂ = the Mult of V | [:COMPLEX, the carrier of b₂:] ) );

theorem :: CLVECT_1:27

for V being ComplexLinearSpace
for W1, W2 being Subspace of V
for x being set st x in W1 & W1 is Subspace of W2 holds
x in W2

proof end;

theorem Th28: :: CLVECT_1:28

for V being ComplexLinearSpace
for W being Subspace of V
for x being object st x in W holds
x in V

proof end;

theorem Th29: :: CLVECT_1:29

for V being ComplexLinearSpace
for W being Subspace of V
for w being VECTOR of W holds w is VECTOR of V

proof end;

theorem :: CLVECT_1:30

for V being ComplexLinearSpace
for W being Subspace of V holds 0. W = 0. V by Def8;

theorem :: CLVECT_1:31

for V being ComplexLinearSpace
for W1, W2 being Subspace of V holds 0. W1 = 0. W2

proof end;

theorem Th32: :: CLVECT_1:32

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u

proof end;

theorem Th33: :: CLVECT_1:33

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex
for W being Subspace of V
for w being VECTOR of W st w = v holds
z * w = z * v

proof end;

theorem Th34: :: CLVECT_1:34

for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V
for w being VECTOR of W st w = v holds
- v = - w

proof end;

theorem Th35: :: CLVECT_1:35

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u

proof end;

Lm3: 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 linearly-closed

proof end;

theorem Th36: :: CLVECT_1:36

for V being ComplexLinearSpace
for W being Subspace of V holds 0. V in W

proof end;

theorem :: CLVECT_1:37

for V being ComplexLinearSpace
for W1, W2 being Subspace of V holds 0. W1 in W2

proof end;

theorem :: CLVECT_1:38

for V being ComplexLinearSpace
for W being Subspace of V holds 0. W in V by Th28, RLVECT_1:1;

theorem Th39: :: CLVECT_1:39

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st u in W & v in W holds
u + v in W

proof end;

theorem Th40: :: CLVECT_1:40

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex
for W being Subspace of V st v in W holds
z * v in W

proof end;

theorem Th41: :: CLVECT_1:41

for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V st v in W holds
- v in W

proof end;

theorem Th42: :: CLVECT_1:42

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st u in W & v in W holds
u - v in W

proof end;

Lm4: now :: thesis: for V being ComplexLinearSpace
for V1 being Subset of V
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:COMPLEX,D:],D
for z being Complex
for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v

let V be ComplexLinearSpace; :: thesis: for V1 being Subset of V
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:COMPLEX,D:],D
for z being Complex
for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let V1 be Subset of V; :: thesis: for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:COMPLEX,D:],D
for z being Complex
for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let D be non empty set ; :: thesis: for d1 being Element of D
for A being BinOp of D
for M being Function of [:COMPLEX,D:],D
for z being Complex
for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let d1 be Element of D; :: thesis: for A being BinOp of D
for M being Function of [:COMPLEX,D:],D
for z being Complex
for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let A be BinOp of D; :: thesis: for M being Function of [:COMPLEX,D:],D
for z being Complex
for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let M be Function of [:COMPLEX,D:],D; :: thesis: for z being Complex
for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let z be Complex; :: thesis: for w being VECTOR of CLSStruct(# D,d1,A,M #)
for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let w be VECTOR of CLSStruct(# D,d1,A,M #); :: thesis: for v being Element of V st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v
let v be Element of V; :: thesis: ( V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v implies z * w = z * v )
assume that
A1: V1 = D and
A2: M = the Mult of V | [:COMPLEX,V1:] and
A3: w = v ; :: thesis: z * w = z * v
z in COMPLEX by XCMPLX_0:def 2;
then [z,w] in [:COMPLEX,V1:] by A1, ZFMISC_1:def 2;
hence z * w = z * v by A3, A2, FUNCT_1:49; :: thesis: verum

end;

theorem Th43: :: CLVECT_1:43

for V being ComplexLinearSpace
for V1 being Subset of V
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:COMPLEX,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:COMPLEX,V1:] holds
CLSStruct(# D,d1,A,M #) is Subspace of V

proof end;

theorem Th44: :: CLVECT_1:44

for V being ComplexLinearSpace holds V is Subspace of V

proof end;

theorem Th45: :: CLVECT_1:45

for V, X being strict ComplexLinearSpace st V is Subspace of X & X is Subspace of V holds
V = X

proof end;

theorem Th46: :: CLVECT_1:46

for V, X, Y being ComplexLinearSpace st V is Subspace of X & X is Subspace of Y holds
V is Subspace of Y

proof end;

theorem Th47: :: CLVECT_1:47

for V being ComplexLinearSpace
for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds
W1 is Subspace of W2

proof end;

theorem :: CLVECT_1:48

for V being ComplexLinearSpace
for W1, W2 being Subspace of V st ( for v being VECTOR of V st v in W1 holds
v in W2 ) holds
W1 is Subspace of W2

proof end;

registration

let V be ComplexLinearSpace;

cluster non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for Subspace of V;

existence
ex b₁ being Subspace of V st b₁ is strict

proof end;

end;

theorem Th49: :: CLVECT_1:49

for V being ComplexLinearSpace
for W1, W2 being strict Subspace of V st the carrier of W1 = the carrier of W2 holds
W1 = W2

proof end;

theorem Th50: :: CLVECT_1:50

for V being ComplexLinearSpace
for W1, W2 being strict Subspace of V st ( for v being VECTOR of V holds
( v in W1 iff v in W2 ) ) holds
W1 = W2

proof end;

theorem :: CLVECT_1:51

for V being strict ComplexLinearSpace
for W being strict Subspace of V st the carrier of W = the carrier of V holds
W = V

proof end;

theorem :: CLVECT_1:52

for V being strict ComplexLinearSpace
for W being strict Subspace of V st ( for v being VECTOR of V holds
( v in W iff v in V ) ) holds
W = V

proof end;

theorem :: CLVECT_1:53

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 linearly-closed by Lm3;

theorem Th54: :: CLVECT_1:54

for V being ComplexLinearSpace
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
ex W being strict Subspace of V st V1 = the carrier of W

proof end;

:: Definition of zero subspace and improper subspace of complex linear space.

definition

let V be ComplexLinearSpace;

func (0). V -> strict Subspace of V means :Def9: :: CLVECT_1:def 9
the carrier of it = {(0. V)};

correctness
existence
ex b₁ being strict Subspace of V st the carrier of b₁ = {(0. V)};
uniqueness
for b₁, b₂ being strict Subspace of V st the carrier of b₁ = {(0. V)} & the carrier of b₂ = {(0. V)} holds
b₁ = b₂;
by Th23, Th49, Th54;

end;

:: deftheorem Def9 defines (0). CLVECT_1:def 9 :
for V being ComplexLinearSpace
for b₂ being strict Subspace of V holds
( b₂ = (0). V iff the carrier of b₂ = {(0. V)} );

definition

let V be ComplexLinearSpace;

func (Omega). V -> strict Subspace of V equals :: CLVECT_1:def 10
CLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);

coherence
CLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) is strict Subspace of V

proof end;

end;

:: deftheorem defines (Omega). CLVECT_1:def 10 :
for V being ComplexLinearSpace holds (Omega). V = CLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);

:: Definitional theorems of zero subspace and improper subspace.

theorem Th55: :: CLVECT_1:55

for V being ComplexLinearSpace
for W being Subspace of V holds (0). W = (0). V

proof end;

theorem Th56: :: CLVECT_1:56

for V being ComplexLinearSpace
for W1, W2 being Subspace of V holds (0). W1 = (0). W2

proof end;

theorem :: CLVECT_1:57

for V being ComplexLinearSpace
for W being Subspace of V holds (0). W is Subspace of V by Th46;

theorem :: CLVECT_1:58

for V being ComplexLinearSpace
for W being Subspace of V holds (0). V is Subspace of W

proof end;

theorem :: CLVECT_1:59

for V being ComplexLinearSpace
for W1, W2 being Subspace of V holds (0). W1 is Subspace of W2

proof end;

theorem :: CLVECT_1:60

for V being strict ComplexLinearSpace holds V is Subspace of (Omega). V ;

:: Introduction of the cosets of subspace.

definition

let V be ComplexLinearSpace;
let v be VECTOR of V;
let W be Subspace of V;

func v + W -> Subset of V equals :: CLVECT_1:def 11
{ (v + u) where u is VECTOR of V : u in W } ;

coherence
{ (v + u) where u is VECTOR of V : u in W } is Subset of V

proof end;

end;

:: deftheorem defines + CLVECT_1:def 11 :
for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V holds v + W = { (v + u) where u is VECTOR of V : u in W } ;

Lm5: for V being ComplexLinearSpace
for W being Subspace of V holds (0. V) + W = the carrier of W

proof end;

definition

let V be ComplexLinearSpace;
let W be Subspace of V;

mode Coset of W -> Subset of V means :Def12: :: CLVECT_1:def 12
ex v being VECTOR of V st it = v + W;

existence
ex b₁ being Subset of V ex v being VECTOR of V st b₁ = v + W

proof end;

end;

:: deftheorem Def12 defines Coset CLVECT_1:def 12 :
for V being ComplexLinearSpace
for W being Subspace of V
for b₃ being Subset of V holds
( b₃ is Coset of W iff ex v being VECTOR of V st b₃ = v + W );

:: Definitional theorems of the cosets.

theorem Th61: :: CLVECT_1:61

for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( 0. V in v + W iff v in W )

proof end;

theorem Th62: :: CLVECT_1:62

for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V holds v in v + W

proof end;

theorem :: CLVECT_1:63

for V being ComplexLinearSpace
for W being Subspace of V holds (0. V) + W = the carrier of W by Lm5;

theorem Th64: :: CLVECT_1:64

for V being ComplexLinearSpace
for v being VECTOR of V holds v + ((0). V) = {v}

proof end;

Lm6: 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 )

proof end;

theorem Th65: :: CLVECT_1:65

for V being ComplexLinearSpace
for v being VECTOR of V holds v + ((Omega). V) = the carrier of V by STRUCT_0:def 5, Lm6;

theorem Th66: :: CLVECT_1:66

for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( 0. V in v + W iff v + W = the carrier of W ) by Th61, Lm6;

theorem :: CLVECT_1:67

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 ) by Lm6;

theorem Th68: :: CLVECT_1:68

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex
for W being Subspace of V st v in W holds
(z * v) + W = the carrier of W

proof end;

theorem Th69: :: CLVECT_1:69

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex
for W being Subspace of V st z <> 0 & (z * v) + W = the carrier of W holds
v in W

proof end;

theorem Th70: :: CLVECT_1:70

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 )

proof end;

theorem Th71: :: CLVECT_1:71

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v + u) + W )

proof end;

theorem :: CLVECT_1:72

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )

proof end;

theorem Th73: :: CLVECT_1:73

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( v in u + W iff u + W = v + W )

proof end;

theorem Th74: :: CLVECT_1:74

for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( v + W = (- v) + W iff v in W )

proof end;

theorem Th75: :: CLVECT_1:75

for V being ComplexLinearSpace
for u, v1, v2 being VECTOR of V
for W being Subspace of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W

proof end;

theorem :: CLVECT_1:76

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st u in v + W & u in (- v) + W holds
v in W by Th75, Th74;

theorem Th77: :: CLVECT_1:77

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex
for W being Subspace of V st z <> 1r & z * v in v + W holds
v in W

proof end;

theorem Th78: :: CLVECT_1:78

for V being ComplexLinearSpace
for v being VECTOR of V
for z being Complex
for W being Subspace of V st v in W holds
z * v in v + W

proof end;

theorem :: CLVECT_1:79

for V being ComplexLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( - v in v + W iff v in W )

proof end;

theorem Th80: :: CLVECT_1:80

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u + v in v + W iff u in W )

proof end;

theorem :: CLVECT_1:81

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( v - u in v + W iff u in W )

proof end;

theorem Th82: :: CLVECT_1:82

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )

proof end;

theorem :: CLVECT_1:83

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )

proof end;

theorem Th84: :: CLVECT_1:84

for V being ComplexLinearSpace
for v1, v2 being VECTOR of V
for W being Subspace of V holds
( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )

proof end;

theorem Th85: :: CLVECT_1:85

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )

proof end;

theorem Th86: :: CLVECT_1:86

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )

proof end;

theorem Th87: :: CLVECT_1:87

for V being ComplexLinearSpace
for v being VECTOR of V
for W1, W2 being strict Subspace of V holds
( v + W1 = v + W2 iff W1 = W2 )

proof end;

theorem Th88: :: CLVECT_1:88

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W1, W2 being strict Subspace of V st v + W1 = u + W2 holds
W1 = W2

proof end;

:: Theorems concerning cosets of subspace
:: regarded as subsets of the carrier.

theorem :: CLVECT_1:89

for V being ComplexLinearSpace
for W being Subspace of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )

proof end;

theorem :: CLVECT_1:90

for V being ComplexLinearSpace
for W1, W2 being strict Subspace of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2

proof end;

theorem :: CLVECT_1:91

for V being ComplexLinearSpace
for v being VECTOR of V holds {v} is Coset of (0). V

proof end;

theorem :: CLVECT_1:92

for V being ComplexLinearSpace
for V1 being Subset of V st V1 is Coset of (0). V holds
ex v being VECTOR of V st V1 = {v}

proof end;

theorem :: CLVECT_1:93

for V being ComplexLinearSpace
for W being Subspace of V holds the carrier of W is Coset of W

proof end;

theorem :: CLVECT_1:94

for V being ComplexLinearSpace holds the carrier of V is Coset of (Omega). V

proof end;

theorem :: CLVECT_1:95

for V being ComplexLinearSpace
for V1 being Subset of V st V1 is Coset of (Omega). V holds
V1 = the carrier of V

proof end;

theorem :: CLVECT_1:96

for V being ComplexLinearSpace
for W being Subspace of V
for C being Coset of W holds
( 0. V in C iff C = the carrier of W )

proof end;

theorem Th97: :: CLVECT_1:97

for V being ComplexLinearSpace
for u being VECTOR of V
for W being Subspace of V
for C being Coset of W holds
( u in C iff C = u + W )

proof end;

theorem :: CLVECT_1:98

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )

proof end;

theorem :: CLVECT_1:99

for V being ComplexLinearSpace
for u, v being VECTOR of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )

proof end;

theorem :: CLVECT_1:100

for V being ComplexLinearSpace
for v1, v2 being VECTOR of V
for W being Subspace of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )

proof end;

theorem :: CLVECT_1:101

for V being ComplexLinearSpace
for u being VECTOR of V
for W being Subspace of V
for B, C being Coset of W st u in B & u in C holds
B = C

proof end;

definition

attr c₁ is strict ;
struct CNORMSTR -> CLSStruct , N-ZeroStr ;
aggr CNORMSTR(# carrier, ZeroF, addF, Mult, normF #) -> CNORMSTR ;

end;

registration

cluster non empty for CNORMSTR ;

existence
not for b₁ being CNORMSTR holds b₁ is empty

proof end;

end;

deffunc H₁( CNORMSTR ) -> Element of the carrier of $1 = 0. $1;

set V = the ComplexLinearSpace;

Lm7: the carrier of ((0). the ComplexLinearSpace) = {(0. the ComplexLinearSpace)}
by Def9;

reconsider niltonil = the carrier of ((0). the ComplexLinearSpace) --> (In (0,REAL)) as Function of the carrier of ((0). the ComplexLinearSpace),REAL ;

0. the ComplexLinearSpace is VECTOR of ((0). the ComplexLinearSpace)
by Lm7, TARSKI:def 1;

then Lm8: niltonil . (0. the ComplexLinearSpace) = 0
by FUNCOP_1:7;

Lm9: for u being VECTOR of ((0). the ComplexLinearSpace)
for z being Complex holds niltonil . (z * u) = |.z.| * (niltonil . u)

proof end;

Lm10: for u, v being VECTOR of ((0). the ComplexLinearSpace) holds niltonil . (u + v) <= (niltonil . u) + (niltonil . v)

proof end;

reconsider X = CNORMSTR(# the carrier of ((0). the ComplexLinearSpace),(0. ((0). the ComplexLinearSpace)), the addF of ((0). the ComplexLinearSpace), the Mult of ((0). the ComplexLinearSpace),niltonil #) as non empty CNORMSTR ;

Lm11: now :: thesis: for x, y being Point of X
for z being Complex holds
( ( ||.x.|| = 0 implies x = H₁(X) ) & ( x = H₁(X) implies ||.x.|| = 0 ) & ||.(z * x).|| = |.z.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )

let x, y be Point of X; :: thesis: for z being Complex holds
( ( ||.x.|| = 0 implies x = H₁(X) ) & ( x = H₁(X) implies ||.x.|| = 0 ) & ||.(z * x).|| = |.z.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
let z be Complex; :: thesis: ( ( ||.x.|| = 0 implies x = H₁(X) ) & ( x = H₁(X) implies ||.x.|| = 0 ) & ||.(z * x).|| = |.z.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
reconsider u = x, w = y as VECTOR of ((0). the ComplexLinearSpace) ;
H₁(X) = 0. the ComplexLinearSpace by Def8;
hence ( ||.x.|| = 0 iff x = H₁(X) ) by Lm7, FUNCOP_1:7, TARSKI:def 1; :: thesis: ( ||.(z * x).|| = |.z.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
z * x = z * u ;
hence ||.(z * x).|| = |.z.| * ||.x.|| by Lm9; :: thesis: ||.(x + y).|| <= ||.x.|| + ||.y.||
x + y = u + w ;
hence ||.(x + y).|| <= ||.x.|| + ||.y.|| by Lm10; :: thesis: verum

end;

definition

let IT be non empty CNORMSTR ;

attr IT is ComplexNormSpace-like means :Def13: :: CLVECT_1:def 13
for x, y being Point of IT
for z being Complex holds
( ||.(z * x).|| = |.z.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| );

end;

:: deftheorem Def13 defines ComplexNormSpace-like CLVECT_1:def 13 :
for IT being non empty CNORMSTR holds
( IT is ComplexNormSpace-like iff for x, y being Point of IT
for z being Complex holds
( ||.(z * x).|| = |.z.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| ) );

registration

cluster non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital strict ComplexNormSpace-like for CNORMSTR ;

existence
ex b₁ being non empty CNORMSTR st
( b₁ is reflexive & b₁ is discerning & b₁ is ComplexNormSpace-like & b₁ is vector-distributive & b₁ is scalar-distributive & b₁ is scalar-associative & b₁ is scalar-unital & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is strict )

proof end;

end;

definition

mode ComplexNormSpace is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR ;

end;

theorem :: CLVECT_1:102

for CNS being ComplexNormSpace holds ||.(0. CNS).|| = 0 ;

theorem Th103: :: CLVECT_1:103

for CNS being ComplexNormSpace
for x being Point of CNS holds ||.(- x).|| = ||.x.||

proof end;

theorem Th104: :: CLVECT_1:104

for CNS being ComplexNormSpace
for x, y being Point of CNS holds ||.(x - y).|| <= ||.x.|| + ||.y.||

proof end;

theorem Th105: :: CLVECT_1:105

for CNS being ComplexNormSpace
for x being Point of CNS holds 0 <= ||.x.||

proof end;

theorem :: CLVECT_1:106

for z1, z2 being Complex
for CNS being ComplexNormSpace
for x, y being Point of CNS holds ||.((z1 * x) + (z2 * y)).|| <= (|.z1.| * ||.x.||) + (|.z2.| * ||.y.||)

proof end;

theorem Th107: :: CLVECT_1:107

for CNS being ComplexNormSpace
for x, y being Point of CNS holds
( ||.(x - y).|| = 0 iff x = y )

proof end;

theorem Th108: :: CLVECT_1:108

for CNS being ComplexNormSpace
for x, y being Point of CNS holds ||.(x - y).|| = ||.(y - x).||

proof end;

theorem Th109: :: CLVECT_1:109

for CNS being ComplexNormSpace
for x, y being Point of CNS holds ||.x.|| - ||.y.|| <= ||.(x - y).||

proof end;

theorem Th110: :: CLVECT_1:110

for CNS being ComplexNormSpace
for x, y being Point of CNS holds |.(||.x.|| - ||.y.||).| <= ||.(x - y).||

proof end;

theorem Th111: :: CLVECT_1:111

for CNS being ComplexNormSpace
for x, y, w being Point of CNS holds ||.(x - w).|| <= ||.(x - y).|| + ||.(y - w).||

proof end;

theorem :: CLVECT_1:112

for CNS being ComplexNormSpace
for x, y being Point of CNS st x <> y holds
||.(x - y).|| <> 0 by Th107;

definition

let CNS be ComplexLinearSpace;
let S be sequence of CNS;
let z be Complex;

func z * S -> sequence of CNS means :Def14: :: CLVECT_1:def 14
for n being Nat holds it . n = z * (S . n);

existence
ex b₁ being sequence of CNS st
for n being Nat holds b₁ . n = z * (S . n)

proof end;

uniqueness
for b₁, b₂ being sequence of CNS st ( for n being Nat holds b₁ . n = z * (S . n) ) & ( for n being Nat holds b₂ . n = z * (S . n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def14 defines * CLVECT_1:def 14 :
for CNS being ComplexLinearSpace
for S being sequence of CNS
for z being Complex
for b₄ being sequence of CNS holds
( b₄ = z * S iff for n being Nat holds b₄ . n = z * (S . n) );

definition

let CNS be ComplexNormSpace;
let S be sequence of CNS;

attr S is convergent means :: CLVECT_1:def 15
ex g being Point of CNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - g).|| < r;

end;

:: deftheorem defines convergent CLVECT_1:def 15 :
for CNS being ComplexNormSpace
for S being sequence of CNS holds
( S is convergent iff ex g being Point of CNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - g).|| < r );

theorem Th113: :: CLVECT_1:113

for CNS being ComplexNormSpace
for S1, S2 being sequence of CNS st S1 is convergent & S2 is convergent holds
S1 + S2 is convergent

proof end;

theorem Th114: :: CLVECT_1:114

for CNS being ComplexNormSpace
for S1, S2 being sequence of CNS st S1 is convergent & S2 is convergent holds
S1 - S2 is convergent

proof end;

theorem Th115: :: CLVECT_1:115

for CNS being ComplexNormSpace
for x being Point of CNS
for S being sequence of CNS st S is convergent holds
S - x is convergent

proof end;

theorem Th116: :: CLVECT_1:116

for z being Complex
for CNS being ComplexNormSpace
for S being sequence of CNS st S is convergent holds
z * S is convergent

proof end;

theorem Th117: :: CLVECT_1:117

for CNS being ComplexNormSpace
for S being sequence of CNS st S is convergent holds
||.S.|| is convergent

proof end;

definition

let CNS be ComplexNormSpace;
let S be sequence of CNS;
assume A1: S is convergent ;

func lim S -> Point of CNS means :Def16: :: CLVECT_1:def 16
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - it).|| < r;

existence
ex b₁ being Point of CNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₁).|| < r by A1;
uniqueness
for b₁, b₂ being Point of CNS st ( for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₁).|| < r ) & ( for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₂).|| < r ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def16 defines lim CLVECT_1:def 16 :
for CNS being ComplexNormSpace
for S being sequence of CNS st S is convergent holds
for b₃ being Point of CNS holds
( b₃ = lim S iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₃).|| < r );

theorem :: CLVECT_1:118

for CNS being ComplexNormSpace
for g being Point of CNS
for S being sequence of CNS st S is convergent & lim S = g holds
( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 )

proof end;

theorem :: CLVECT_1:119

for CNS being ComplexNormSpace
for S1, S2 being sequence of CNS st S1 is convergent & S2 is convergent holds
lim (S1 + S2) = (lim S1) + (lim S2)

proof end;

theorem :: CLVECT_1:120

for CNS being ComplexNormSpace
for S1, S2 being sequence of CNS st S1 is convergent & S2 is convergent holds
lim (S1 - S2) = (lim S1) - (lim S2)

proof end;

theorem :: CLVECT_1:121

for CNS being ComplexNormSpace
for x being Point of CNS
for S being sequence of CNS st S is convergent holds
lim (S - x) = (lim S) - x

proof end;

theorem :: CLVECT_1:122

for z being Complex
for CNS being ComplexNormSpace
for S being sequence of CNS st S is convergent holds
lim (z * S) = z * (lim S)

proof end;

theorem :: CLVECT_1:123

for z being Complex
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:COMPLEX,D:],D
for V being ComplexLinearSpace
for V1 being Subset of V
for v being VECTOR of V
for w being VECTOR of CLSStruct(# D,d1,A,M #) st V1 = D & M = the Mult of V | [:COMPLEX,V1:] & w = v holds
z * w = z * v by Lm4;