:: VFUNCT_2 semantic presentation

definition

let M be non empty set ;
let V be ComplexNormSpace;
let f1, f2 be PartFunc of M,the carrier of V;
deffunc H₁( Element of M) -> Element of the carrier of V = (f1 /. $1) + (f2 /. $1);
deffunc H₂( Element of M) -> Element of the carrier of V = (f1 /. $1) - (f2 /. $1);
defpred S₁[ set ] means $1 in (dom f1) /\ (dom f2);
set X = (dom f1) /\ (dom f2);
func f1 + f2 -> PartFunc of M,the carrier of V means :Def1: :: VFUNCT_2:def 1
( dom it = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom it holds
it /. c = (f1 /. c) + (f2 /. c) ) );
existence

proof end;

uniqueness

proof end;

func f1 - f2 -> PartFunc of M,the carrier of V means :Def2: :: VFUNCT_2:def 2
( dom it = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom it holds
it /. c = (f1 /. c) - (f2 /. c) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines + VFUNCT_2:def 1 :

:: deftheorem Def2 defines - VFUNCT_2:def 2 :

definition

let M be non empty set ;
let V be ComplexNormSpace;
let f1 be PartFunc of M, COMPLEX ;
let f2 be PartFunc of M,the carrier of V;
deffunc H₁( Element of M) -> Element of the carrier of V = (f1 /. $1) * (f2 /. $1);
defpred S₁[ set ] means $1 in (dom f1) /\ (dom f2);
set X = (dom f1) /\ (dom f2);
func f1 (#) f2 -> PartFunc of M,the carrier of V means :Def3: :: VFUNCT_2:def 3
( dom it = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom it holds
it /. c = (f1 /. c) * (f2 /. c) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines (#) VFUNCT_2:def 3 :

definition

let X be non empty set ;
let V be ComplexNormSpace;
let f be PartFunc of X,the carrier of V;
let z be Complex;
deffunc H₁( Element of X) -> Element of the carrier of V = z * (f /. $1);
defpred S₁[ set ] means $1 in dom f;
set M = dom f;
func z (#) f -> PartFunc of X,the carrier of V means :Def4: :: VFUNCT_2:def 4
( dom it = dom f & ( for x being Element of X st x in dom it holds
it /. x = z * (f /. x) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines (#) VFUNCT_2:def 4 :

definition

let X be non empty set ;
let V be ComplexNormSpace;
let f be PartFunc of X,the carrier of V;
deffunc H₁( Element of X) -> Element of REAL = ||.(f /. $1).||;
deffunc H₂( Element of X) -> Element of the carrier of V = - (f /. $1);
defpred S₁[ set ] means $1 in dom f;
set M = dom f;
func ||.f.|| -> PartFunc of X, REAL means :Def5: :: VFUNCT_2:def 5
( dom it = dom f & ( for x being Element of X st x in dom it holds
it . x = ||.(f /. x).|| ) );
existence

proof end;

uniqueness

proof end;

func - f -> PartFunc of X,the carrier of V means :Def6: :: VFUNCT_2:def 6
( dom it = dom f & ( for x being Element of X st x in dom it holds
it /. x = - (f /. x) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines ||. VFUNCT_2:def 5 :

:: deftheorem Def6 defines - VFUNCT_2:def 6 :

theorem :: VFUNCT_2:1

for M being non empty set
for V being ComplexNormSpace
for f1 being PartFunc of M, COMPLEX
for f2 being PartFunc of M,the carrier of V holds (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) = ((dom f1) \ (f1 " {0})) /\ ((dom f2) \ (f2 " {(0. V)}))

proof end;

theorem :: VFUNCT_2:2

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds
( ||.f.|| " {0} = f " {(0. V)} & (- f) " {(0. V)} = f " {(0. V)} )

proof end;

theorem :: VFUNCT_2:3

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex st z <> 0c holds
(z (#) f) " {(0. V)} = f " {(0. V)}

proof end;

theorem :: VFUNCT_2:4

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 + f2 = f2 + f1

proof end;

theorem :: VFUNCT_2:5

for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds (f1 + f2) + f3 = f1 + (f2 + f3)

proof end;

theorem :: VFUNCT_2:6

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M, COMPLEX
for f3 being PartFunc of M,the carrier of V holds (f1 (#) f2) (#) f3 = f1 (#) (f2 (#) f3)

proof end;

theorem :: VFUNCT_2:7

for M being non empty set
for V being ComplexNormSpace
for f3 being PartFunc of M,the carrier of V
for f1, f2 being PartFunc of M, COMPLEX holds (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3)

proof end;

theorem :: VFUNCT_2:8

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for f3 being PartFunc of M, COMPLEX holds f3 (#) (f1 + f2) = (f3 (#) f1) + (f3 (#) f2)

proof end;

theorem :: VFUNCT_2:9

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for z being Complex
for f1 being PartFunc of M, COMPLEX holds z (#) (f1 (#) f2) = (z (#) f1) (#) f2

proof end;

theorem :: VFUNCT_2:10

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for z being Complex
for f1 being PartFunc of M, COMPLEX holds z (#) (f1 (#) f2) = f1 (#) (z (#) f2)

proof end;

theorem :: VFUNCT_2:11

for M being non empty set
for V being ComplexNormSpace
for f3 being PartFunc of M,the carrier of V
for f1, f2 being PartFunc of M, COMPLEX holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)

proof end;

theorem :: VFUNCT_2:12

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for f3 being PartFunc of M, COMPLEX holds (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2)

proof end;

theorem :: VFUNCT_2:13

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for z being Complex holds z (#) (f1 + f2) = (z (#) f1) + (z (#) f2)

proof end;

theorem Th14: :: VFUNCT_2:14

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z1, z2 being Complex holds (z1 * z2) (#) f = z1 (#) (z2 (#) f)

proof end;

theorem :: VFUNCT_2:15

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for z being Complex holds z (#) (f1 - f2) = (z (#) f1) - (z (#) f2)

proof end;

theorem :: VFUNCT_2:16

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 - f2 = (- 1r ) (#) (f2 - f1)

proof end;

theorem :: VFUNCT_2:17

for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds f1 - (f2 + f3) = (f1 - f2) - f3

proof end;

theorem Th18: :: VFUNCT_2:18

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds 1r (#) f = f

proof end;

theorem :: VFUNCT_2:19

for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds f1 - (f2 - f3) = (f1 - f2) + f3

proof end;

theorem :: VFUNCT_2:20

for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds f1 + (f2 - f3) = (f1 + f2) - f3

proof end;

theorem :: VFUNCT_2:21

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for f1 being PartFunc of M, COMPLEX holds ||.(f1 (#) f2).|| = |.f1.| (#) ||.f2.||

proof end;

theorem :: VFUNCT_2:22

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex holds ||.(z (#) f).|| = |.z.| (#) ||.f.||

proof end;

theorem Th23: :: VFUNCT_2:23

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds - f = (- 1r ) (#) f

proof end;

theorem Th24: :: VFUNCT_2:24

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds - (- f) = f

proof end;

theorem Th25: :: VFUNCT_2:25

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 - f2 = f1 + (- f2)

proof end;

theorem :: VFUNCT_2:26

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 - (- f2) = f1 + f2

proof end;

theorem Th27: :: VFUNCT_2:27

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X being set holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )

proof end;

theorem :: VFUNCT_2:28

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for X being set
for f1 being PartFunc of M, COMPLEX holds
( (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) & (f1 (#) f2) | X = (f1 | X) (#) f2 & (f1 (#) f2) | X = f1 (#) (f2 | X) )

proof end;

theorem Th29: :: VFUNCT_2:29

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for X being set holds
( (- f) | X = - (f | X) & ||.f.|| | X = ||.(f | X).|| )

proof end;

theorem :: VFUNCT_2:30

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X being set holds
( (f1 - f2) | X = (f1 | X) - (f2 | X) & (f1 - f2) | X = (f1 | X) - f2 & (f1 - f2) | X = f1 - (f2 | X) )

proof end;

theorem :: VFUNCT_2:31

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for X being set holds (z (#) f) | X = z (#) (f | X)

proof end;

theorem Th32: :: VFUNCT_2:32

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds
( ( f1 is total & f2 is total implies f1 + f2 is total ) & ( f1 + f2 is total implies ( f1 is total & f2 is total ) ) & ( f1 is total & f2 is total implies f1 - f2 is total ) & ( f1 - f2 is total implies ( f1 is total & f2 is total ) ) )

proof end;

theorem Th33: :: VFUNCT_2:33

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for f1 being PartFunc of M, COMPLEX holds
( ( f1 is total & f2 is total ) iff f1 (#) f2 is total )

proof end;

theorem Th34: :: VFUNCT_2:34

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex holds
( f is total iff z (#) f is total )

proof end;

theorem Th35: :: VFUNCT_2:35

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds
( f is total iff - f is total )

proof end;

theorem Th36: :: VFUNCT_2:36

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds
( f is total iff ||.f.|| is total )

proof end;

theorem :: VFUNCT_2:37

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for x being Element of M st f1 is total & f2 is total holds
( (f1 + f2) /. x = (f1 /. x) + (f2 /. x) & (f1 - f2) /. x = (f1 /. x) - (f2 /. x) )

proof end;

theorem :: VFUNCT_2:38

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for f1 being PartFunc of M, COMPLEX
for x being Element of M st f1 is total & f2 is total holds
(f1 (#) f2) /. x = (f1 /. x) * (f2 /. x)

proof end;

theorem :: VFUNCT_2:39

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for x being Element of M st f is total holds
(z (#) f) /. x = z * (f /. x)

proof end;

theorem :: VFUNCT_2:40

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for x being Element of M st f is total holds
( (- f) /. x = - (f /. x) & ||.f.|| . x = ||.(f /. x).|| )

proof end;

definition

let M be non empty set ;
let V be ComplexNormSpace;
let f be PartFunc of M,the carrier of V;
let Y be set ;
pred f is_bounded_on Y means :Def7: :: VFUNCT_2:def 7
ex r being Real st
for x being Element of M st x in Y /\ (dom f) holds
||.(f /. x).|| <= r;

end;

:: deftheorem Def7 defines is_bounded_on VFUNCT_2:def 7 :

theorem :: VFUNCT_2:41

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y, X being set st Y c= X & f is_bounded_on X holds
f is_bounded_on Y

proof end;

theorem :: VFUNCT_2:42

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for X being set st X misses dom f holds
f is_bounded_on X

proof end;

theorem :: VFUNCT_2:43

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set holds 0c (#) f is_bounded_on Y

proof end;

theorem Th44: :: VFUNCT_2:44

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for Y being set st f is_bounded_on Y holds
z (#) f is_bounded_on Y

proof end;

theorem Th45: :: VFUNCT_2:45

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_bounded_on Y holds
( ||.f.|| is_bounded_on Y & - f is_bounded_on Y )

proof end;

theorem Th46: :: VFUNCT_2:46

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 + f2 is_bounded_on X /\ Y

proof end;

theorem :: VFUNCT_2:47

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for X, Y being set
for f1 being PartFunc of M, COMPLEX st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 (#) f2 is_bounded_on X /\ Y

proof end;

theorem :: VFUNCT_2:48

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 - f2 is_bounded_on X /\ Y

proof end;

theorem :: VFUNCT_2:49

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for X, Y being set st f is_bounded_on X & f is_bounded_on Y holds
f is_bounded_on X \/ Y

proof end;

theorem :: VFUNCT_2:50

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_constant_on X & f2 is_constant_on Y holds
( f1 + f2 is_constant_on X /\ Y & f1 - f2 is_constant_on X /\ Y )

proof end;

theorem :: VFUNCT_2:51

for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for X, Y being set
for f1 being PartFunc of M, COMPLEX st f1 is_constant_on X & f2 is_constant_on Y holds
f1 (#) f2 is_constant_on X /\ Y

proof end;

theorem Th52: :: VFUNCT_2:52

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for Y being set st f is_constant_on Y holds
z (#) f is_constant_on Y

proof end;

theorem Th53: :: VFUNCT_2:53

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_constant_on Y holds
( ||.f.|| is_constant_on Y & - f is_constant_on Y )

proof end;

theorem Th54: :: VFUNCT_2:54

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_constant_on Y holds
f is_bounded_on Y

proof end;

theorem :: VFUNCT_2:55

for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_constant_on Y holds
( ( for z being Complex holds z (#) f is_bounded_on Y ) & - f is_bounded_on Y & ||.f.|| is_bounded_on Y )

proof end;

theorem :: VFUNCT_2:56

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_constant_on Y holds
f1 + f2 is_bounded_on X /\ Y

proof end;

theorem :: VFUNCT_2:57

for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_constant_on Y holds
( f1 - f2 is_bounded_on X /\ Y & f2 - f1 is_bounded_on X /\ Y )

proof end;