begin
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:
(
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
ex b1 being PartFunc of M,the carrier of V st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of M,the carrier of V st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b2 holds
b2 /. c = (f1 /. c) + (f2 /. c) ) holds
b1 = b2
func f1 - f2 -> PartFunc of
M,the
carrier of
V means :
Def2:
(
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
ex b1 being PartFunc of M,the carrier of V st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of M,the carrier of V st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b2 holds
b2 /. c = (f1 /. c) - (f2 /. c) ) holds
b1 = b2
end;
:: deftheorem Def1 defines + VFUNCT_2:def 1 :
for M being non empty set
for V being ComplexNormSpace
for f1, f2, b5 being PartFunc of M,the carrier of V holds
( b5 = f1 + f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b5 holds
b5 /. c = (f1 /. c) + (f2 /. c) ) ) );
:: deftheorem Def2 defines - VFUNCT_2:def 2 :
for M being non empty set
for V being ComplexNormSpace
for f1, f2, b5 being PartFunc of M,the carrier of V holds
( b5 = f1 - f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b5 holds
b5 /. c = (f1 /. c) - (f2 /. c) ) ) );
:: deftheorem Def3 defines (#) VFUNCT_2:def 3 :
for M being non empty set
for V being ComplexNormSpace
for f1 being PartFunc of M,COMPLEX
for f2, b5 being PartFunc of M,the carrier of V holds
( b5 = f1 (#) f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b5 holds
b5 /. c = (f1 /. c) * (f2 /. c) ) ) );
:: deftheorem Def4 defines (#) VFUNCT_2:def 4 :
for X being non empty set
for V being ComplexNormSpace
for f being PartFunc of X,the carrier of V
for z being Complex
for b5 being PartFunc of X,the carrier of V holds
( b5 = z (#) f iff ( dom b5 = dom f & ( for x being Element of X st x in dom b5 holds
b5 /. x = z * (f /. x) ) ) );
:: deftheorem VFUNCT_2:def 5 :
canceled;
:: deftheorem Def6 defines - VFUNCT_2:def 6 :
for X being non empty set
for V being ComplexNormSpace
for f, b4 being PartFunc of X,the carrier of V holds
( b4 = - f iff ( dom b4 = dom f & ( for x being Element of X st x in dom b4 holds
b4 /. x = - (f /. x) ) ) );
theorem
theorem
theorem
theorem aa:
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem Th14:
theorem
theorem
theorem
theorem Th18:
theorem
theorem
theorem
theorem
theorem Th23:
theorem Th24:
theorem Th25:
theorem
theorem Th27:
theorem
theorem Th29:
theorem
theorem
theorem Th32:
theorem Th33:
theorem Th34:
theorem Th35:
theorem Th36:
theorem
theorem
theorem
theorem
:: deftheorem Def7 defines is_bounded_on VFUNCT_2:def 7 :
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
( f is_bounded_on Y iff ex r being Real st
for x being Element of M st x in Y /\ (dom f) holds
||.(f /. x).|| <= r );
theorem
theorem
theorem
theorem Th44:
theorem Th45:
theorem Th46:
theorem
theorem
theorem
theorem
theorem
theorem Th52:
theorem Th53:
theorem Th54:
theorem
theorem
theorem