Article LOPBAN_5, MML version 4.99.1005
:: LOPBAN_5:th 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
st b1 is bounded & 0 <= b2
holds lim_inf (b2 (#) b1) = b2 * lim_inf b1;
:: LOPBAN_5:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
st b1 is bounded & 0 <= b2
holds lim_sup (b2 (#) b1) = b2 * lim_sup b1;
:: LOPBAN_5:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
cluster MetricSpaceNorm a1 -> non empty Reflexive discerning symmetric triangle complete;
end;
:: LOPBAN_5:funcnot 1 => LOPBAN_5:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
let a2 be Element of the carrier of a1;
let a3 be real set;
func Ball(A2,A3) -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| < a3};
end;
:: LOPBAN_5:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Element of the carrier of b1
for b3 being real set holds
Ball(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| < b3};
:: LOPBAN_5:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st union proj2 b2 = the carrier of b1 &
(for b3 being Element of NAT holds
b2 . b3 is closed(b1))
holds ex b3 being Element of NAT st
ex b4 being Element of REAL st
ex b5 being Element of the carrier of b1 st
0 < b4 & Ball(b5,b4) c= b2 . b3;
:: LOPBAN_5:th 4
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2 holds
b3 is_Lipschitzian_on the carrier of b1 &
b3 is_continuous_on the carrier of b1 &
(for b4 being Element of the carrier of b1 holds
b3 is_continuous_in b4);
:: LOPBAN_5:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of bool the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
st for b4 being Element of the carrier of b1 holds
ex b5 being real set st
0 <= b5 &
(for b6 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
st b6 in b3
holds ||.b6 . b4.|| <= b5)
holds ex b4 being real set st
0 <= b4 &
(for b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
st b5 in b3
holds ||.b5.|| <= b4);
:: LOPBAN_5:funcnot 2 => LOPBAN_5:func 2
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2);
let a4 be Element of the carrier of a1;
func A3 # A4 -> Function-like quasi_total Relation of NAT,the carrier of a2 means
for b1 being Element of NAT holds
it . b1 = (a3 . b1) . a4;
end;
:: LOPBAN_5:def 2
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b5 = b3 # b4
iff
for b6 being Element of NAT holds
b5 . b6 = (b3 . b6) . b4;
:: LOPBAN_5:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st for b5 being Element of the carrier of b1 holds
b3 # b5 is convergent(b2) & b4 . b5 = lim (b3 # b5)
holds b4 is Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2 &
(for b5 being Element of the carrier of b1 holds
||.b4 . b5.|| <= (lim_inf ||.b3.||) * ||.b5.||) &
(for b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
st b5 = b4
holds ||.b5.|| <= lim_inf ||.b3.||);
:: LOPBAN_5:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Element of bool the carrier of LinearTopSpaceNorm b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b3)
st b2 is dense(LinearTopSpaceNorm b1) &
(for b5 being Element of the carrier of b1
st b5 in b2
holds b4 # b5 is convergent(b3)) &
(for b5 being Element of the carrier of b1 holds
ex b6 being real set st
0 <= b6 &
(for b7 being Element of NAT holds
||.(b4 # b5) . b7.|| <= b6))
for b5 being Element of the carrier of b1 holds
b4 # b5 is convergent(b3);
:: LOPBAN_5:th 8
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b3 being Element of bool the carrier of LinearTopSpaceNorm b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
st b3 is dense(LinearTopSpaceNorm b1) &
(for b5 being Element of the carrier of b1
st b5 in b3
holds b4 # b5 is convergent(b2)) &
(for b5 being Element of the carrier of b1 holds
ex b6 being real set st
0 <= b6 &
(for b7 being Element of NAT holds
||.(b4 # b5) . b7.|| <= b6))
holds ex b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2) st
(for b6 being Element of the carrier of b1 holds
b4 # b6 is convergent(b2) &
b5 . b6 = lim (b4 # b6) &
||.b5 . b6.|| <= (lim_inf ||.b4.||) * ||.b6.||) &
||.b5.|| <= lim_inf ||.b4.||;