Article CLVECT_3, MML version 4.99.1005
:: CLVECT_3:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) + Partial_Sums b3 = Partial_Sums (b2 + b3);
:: CLVECT_3:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) - Partial_Sums b3 = Partial_Sums (b2 - b3);
:: CLVECT_3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of COMPLEX holds
Partial_Sums (b3 * b2) = b3 * Partial_Sums b2;
:: CLVECT_3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Partial_Sums - b2 = - Partial_Sums b2;
:: CLVECT_3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4, b5 being Element of COMPLEX holds
(b4 * Partial_Sums b2) + (b5 * Partial_Sums b3) = Partial_Sums ((b4 * b2) + (b5 * b3));
:: CLVECT_3:attrnot 1 => CLVECT_3:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is summable means
Partial_Sums a2 is convergent(a1);
end;
:: CLVECT_3:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is summable
it is sufficient to prove
thus Partial_Sums a2 is convergent(a1);
:: CLVECT_3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is summable(b1)
iff
Partial_Sums b2 is convergent(b1);
:: CLVECT_3:funcnot 1 => CLVECT_3:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
func Sum A2 -> Element of the carrier of a1 equals
lim Partial_Sums a2;
end;
:: CLVECT_3:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Sum b2 = lim Partial_Sums b2;
:: CLVECT_3:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1) & b3 is summable(b1)
holds b2 + b3 is summable(b1) &
Sum (b2 + b3) = (Sum b2) + Sum b3;
:: CLVECT_3:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1) & b3 is summable(b1)
holds b2 - b3 is summable(b1) &
Sum (b2 - b3) = (Sum b2) - Sum b3;
:: CLVECT_3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of COMPLEX
st b2 is summable(b1)
holds b3 * b2 is summable(b1) & Sum (b3 * b2) = b3 * Sum b2;
:: CLVECT_3:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1)
holds b2 is convergent(b1) & lim b2 = 0. b1;
:: CLVECT_3:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b1 is Hilbert
holds b2 is summable(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5, b6 being Element of NAT
st b4 <= b5 & b4 <= b6
holds ||.((Partial_Sums b2) . b5) - ((Partial_Sums b2) . b6).|| < b3;
:: CLVECT_3:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1)
holds Partial_Sums b2 is bounded(b1);
:: CLVECT_3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b4 being Element of NAT holds
b2 . b4 = b3 . 0
holds Partial_Sums (b3 ^\ 1) = ((Partial_Sums b3) ^\ 1) - b2;
:: CLVECT_3:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1)
for b3 being Element of NAT holds
b2 ^\ b3 is summable(b1);
:: CLVECT_3:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st ex b3 being Element of NAT st
b2 ^\ b3 is summable(b1)
holds b2 is summable(b1);
:: CLVECT_3:funcnot 2 => CLVECT_3:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of NAT;
func Sum(A2,A3) -> Element of the carrier of a1 equals
(Partial_Sums a2) . a3;
end;
:: CLVECT_3:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
Sum(b2,b3) = (Partial_Sums b2) . b3;
:: CLVECT_3:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Sum(b2,0) = b2 . 0;
:: CLVECT_3:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Sum(b2,1) = (Sum(b2,0)) + (b2 . 1);
:: CLVECT_3:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Sum(b2,1) = (b2 . 0) + (b2 . 1);
:: CLVECT_3:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
Sum(b2,b3 + 1) = (Sum(b2,b3)) + (b2 . (b3 + 1));
:: CLVECT_3:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
b2 . (b3 + 1) = (Sum(b2,b3 + 1)) - Sum(b2,b3);
:: CLVECT_3:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 . 1 = (Sum(b2,1)) - Sum(b2,0);
:: CLVECT_3:funcnot 3 => CLVECT_3:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3, a4 be Element of NAT;
func Sum(A2,A3,A4) -> Element of the carrier of a1 equals
(Sum(a2,a3)) - Sum(a2,a4);
end;
:: CLVECT_3:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
Sum(b2,b3,b4) = (Sum(b2,b3)) - Sum(b2,b4);
:: CLVECT_3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Sum(b2,1,0) = b2 . 1;
:: CLVECT_3:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
Sum(b2,b3 + 1,b3) = b2 . (b3 + 1);
:: CLVECT_3:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b1 is Hilbert
holds b2 is summable(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5, b6 being Element of NAT
st b4 <= b5 & b4 <= b6
holds ||.(Sum(b2,b5)) - Sum(b2,b6).|| < b3;
:: CLVECT_3:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b1 is Hilbert
holds b2 is summable(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5, b6 being Element of NAT
st b4 <= b5 & b4 <= b6
holds ||.Sum(b2,b5,b6).|| < b3;
:: CLVECT_3:funcnot 4 => CLVECT_3:func 4
definition
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
let a2 be Element of NAT;
func Sum(A1,A2) -> Element of COMPLEX equals
(Partial_Sums a1) . a2;
end;
:: CLVECT_3:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT holds
Sum(b1,b2) = (Partial_Sums b1) . b2;
:: CLVECT_3:funcnot 5 => CLVECT_3:func 5
definition
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
let a2, a3 be Element of NAT;
func Sum(A1,A2,A3) -> Element of COMPLEX equals
(Sum(a1,a2)) - Sum(a1,a3);
end;
:: CLVECT_3:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2, b3 being Element of NAT holds
Sum(b1,b2,b3) = (Sum(b1,b2)) - Sum(b1,b3);
:: CLVECT_3:attrnot 2 => CLVECT_3:attr 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is absolutely_summable means
||.a2.|| is summable;
end;
:: CLVECT_3:dfs 7
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is absolutely_summable
it is sufficient to prove
thus ||.a2.|| is summable;
:: CLVECT_3:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is absolutely_summable(b1)
iff
||.b2.|| is summable;
:: CLVECT_3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is absolutely_summable(b1) & b3 is absolutely_summable(b1)
holds b2 + b3 is absolutely_summable(b1);
:: CLVECT_3:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of COMPLEX
st b2 is absolutely_summable(b1)
holds b3 * b2 is absolutely_summable(b1);
:: CLVECT_3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
||.b2.|| . b4 <= b3 . b4) &
b3 is summable
holds b2 is absolutely_summable(b1);
:: CLVECT_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b2 . b4 <> 0. b1 &
b3 . b4 = ||.b2 . (b4 + 1).|| / ||.b2 . b4.||) &
b3 is convergent &
lim b3 < 1
holds b2 is absolutely_summable(b1);
:: CLVECT_3:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL
st 0 < b3 &
(ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds b3 <= ||.b2 . b5.||) &
b2 is convergent(b1)
holds lim b2 <> 0. b1;
:: CLVECT_3:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st (for b3 being Element of NAT holds
b2 . b3 <> 0. b1) &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds 1 <= ||.b2 . (b4 + 1).|| / ||.b2 . b4.||)
holds b2 is not summable(b1);
:: CLVECT_3:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b2 . b4 <> 0. b1) &
(for b4 being Element of NAT holds
b3 . b4 = ||.b2 . (b4 + 1).|| / ||.b2 . b4.||) &
b3 is convergent &
1 < lim b3
holds b2 is not summable(b1);
:: CLVECT_3:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b3 . b4 = b4 -root ||.b2 . b4.||) &
b3 is convergent &
lim b3 < 1
holds b2 is absolutely_summable(b1);
:: CLVECT_3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b3 . b4 = b4 -root (||.b2.|| . b4)) &
(ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds 1 <= b3 . b5)
holds b2 is not summable(b1);
:: CLVECT_3:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b3 . b4 = b4 -root (||.b2.|| . b4)) &
b3 is convergent &
1 < lim b3
holds b2 is not summable(b1);
:: CLVECT_3:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Partial_Sums ||.b2.|| is non-decreasing;
:: CLVECT_3:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
0 <= (Partial_Sums ||.b2.||) . b3;
:: CLVECT_3:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
||.(Partial_Sums b2) . b3.|| <= (Partial_Sums ||.b2.||) . b3;
:: CLVECT_3:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
||.Sum(b2,b3).|| <= Sum(||.b2.||,b3);
:: CLVECT_3:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
||.((Partial_Sums b2) . b4) - ((Partial_Sums b2) . b3).|| <= abs (((Partial_Sums ||.b2.||) . b4) - ((Partial_Sums ||.b2.||) . b3));
:: CLVECT_3:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
||.(Sum(b2,b4)) - Sum(b2,b3).|| <= abs ((Sum(||.b2.||,b4)) - Sum(||.b2.||,b3));
:: CLVECT_3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
||.Sum(b2,b4,b3).|| <= abs Sum(||.b2.||,b4,b3);
:: CLVECT_3:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b1 is Hilbert & b2 is absolutely_summable(b1)
holds b2 is summable(b1);
:: CLVECT_3:funcnot 6 => CLVECT_3:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Function-like quasi_total Relation of NAT,COMPLEX;
func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
for b1 being Element of NAT holds
it . b1 = (a3 . b1) * (a2 . b1);
end;
:: CLVECT_3:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b4 = b3 * b2
iff
for b5 being Element of NAT holds
b4 . b5 = (b3 . b5) * (b2 . b5);
:: CLVECT_3:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,COMPLEX holds
b4 * (b2 + b3) = (b4 * b2) + (b4 * b3);
:: CLVECT_3:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2);
:: CLVECT_3:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,COMPLEX holds
(b3 (#) b4) * b2 = b3 * (b4 * b2);
:: CLVECT_3:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
for b4 being Element of COMPLEX holds
(b4 (#) b3) * b2 = b4 * (b3 * b2);
:: CLVECT_3:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
b3 * - b2 = (- b3) * b2;
:: CLVECT_3:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
st b3 is convergent & b2 is convergent(b1)
holds b3 * b2 is convergent(b1);
:: CLVECT_3:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
st b3 is bounded & b2 is bounded(b1)
holds b3 * b2 is bounded(b1);
:: CLVECT_3:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
st b3 is convergent & b2 is convergent(b1)
holds b3 * b2 is convergent(b1) &
lim (b3 * b2) = (lim b3) * lim b2;
:: CLVECT_3:attrnot 3 => CLVECT_3:attr 3
definition
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
attr a1 is Cauchy means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3, b4 being Element of NAT
st b2 <= b3 & b2 <= b4
holds |.(a1 . b3) - (a1 . b4).| < b1;
end;
:: CLVECT_3:dfs 9
definiens
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
To prove
a1 is Cauchy
it is sufficient to prove
thus for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3, b4 being Element of NAT
st b2 <= b3 & b2 <= b4
holds |.(a1 . b3) - (a1 . b4).| < b1;
:: CLVECT_3:def 10
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
b1 is Cauchy
iff
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4, b5 being Element of NAT
st b3 <= b4 & b3 <= b5
holds |.(b1 . b4) - (b1 . b5).| < b2;
:: CLVECT_3:prednot 1 => CLVECT_3:attr 3
notation
let a1 be Function-like quasi_total Relation of NAT,COMPLEX;
synonym a1 is_Cauchy_sequence for Cauchy;
end;
:: CLVECT_3:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
st b1 is Hilbert & b2 is Cauchy(b1) & b3 is Cauchy
holds b3 * b2 is Cauchy(b1);
:: CLVECT_3:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
for b4 being Element of NAT holds
(Partial_Sums ((b3 - (b3 ^\ 1)) * Partial_Sums b2)) . b4 = ((Partial_Sums (b3 * b2)) . (b4 + 1)) - ((b3 * Partial_Sums b2) . (b4 + 1));
:: CLVECT_3:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
for b4 being Element of NAT holds
(Partial_Sums (b3 * b2)) . (b4 + 1) = ((b3 * Partial_Sums b2) . (b4 + 1)) - ((Partial_Sums (((b3 ^\ 1) - b3) * Partial_Sums b2)) . b4);
:: CLVECT_3:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
for b4 being Element of NAT holds
Sum(b3 * b2,b4 + 1) = ((b3 * Partial_Sums b2) . (b4 + 1)) - Sum(((b3 ^\ 1) - b3) * Partial_Sums b2,b4);