Article RSSPACE, MML version 4.99.1005
:: RSSPACE:funcnot 1 => RSSPACE:func 1
definition
func the_set_of_RealSequences -> non empty set means
for b1 being set holds
b1 in it
iff
b1 is Function-like quasi_total Relation of NAT,REAL;
end;
:: RSSPACE:def 1
theorem
for b1 being non empty set holds
b1 = the_set_of_RealSequences
iff
for b2 being set holds
b2 in b1
iff
b2 is Function-like quasi_total Relation of NAT,REAL;
:: RSSPACE:funcnot 2 => RSSPACE:func 2
definition
let a1 be set;
assume a1 in the_set_of_RealSequences;
func seq_id A1 -> Function-like quasi_total Relation of NAT,REAL equals
a1;
end;
:: RSSPACE:def 2
theorem
for b1 being set
st b1 in the_set_of_RealSequences
holds seq_id b1 = b1;
:: RSSPACE:funcnot 3 => RSSPACE:func 3
definition
let a1 be set;
assume a1 in REAL;
func R_id A1 -> Element of REAL equals
a1;
end;
:: RSSPACE:def 3
theorem
for b1 being set
st b1 in REAL
holds R_id b1 = b1;
:: RSSPACE:th 1
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences st
(for b2, b3 being Element of the_set_of_RealSequences holds
b1 .(b2,b3) = (seq_id b2) + seq_id b3) &
b1 is commutative(the_set_of_RealSequences) &
b1 is associative(the_set_of_RealSequences);
:: RSSPACE:th 2
theorem
ex b1 being Function-like quasi_total Relation of [:REAL,the_set_of_RealSequences:],the_set_of_RealSequences st
for b2, b3 being set
st b2 in REAL & b3 in the_set_of_RealSequences
holds b1 .(b2,b3) = (R_id b2) (#) seq_id b3;
:: RSSPACE:funcnot 4 => RSSPACE:func 4
definition
func l_add -> Function-like quasi_total Relation of [:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences means
for b1, b2 being Element of the_set_of_RealSequences holds
it .(b1,b2) = (seq_id b1) + seq_id b2;
end;
:: RSSPACE:def 4
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences holds
b1 = l_add
iff
for b2, b3 being Element of the_set_of_RealSequences holds
b1 .(b2,b3) = (seq_id b2) + seq_id b3;
:: RSSPACE:funcnot 5 => RSSPACE:func 5
definition
func l_mult -> Function-like quasi_total Relation of [:REAL,the_set_of_RealSequences:],the_set_of_RealSequences means
for b1, b2 being set
st b1 in REAL & b2 in the_set_of_RealSequences
holds it .(b1,b2) = (R_id b1) (#) seq_id b2;
end;
:: RSSPACE:def 5
theorem
for b1 being Function-like quasi_total Relation of [:REAL,the_set_of_RealSequences:],the_set_of_RealSequences holds
b1 = l_mult
iff
for b2, b3 being set
st b2 in REAL & b3 in the_set_of_RealSequences
holds b1 .(b2,b3) = (R_id b2) (#) seq_id b3;
:: RSSPACE:funcnot 6 => RSSPACE:func 6
definition
func Zeroseq -> Element of the_set_of_RealSequences means
for b1 being Element of NAT holds
(seq_id it) . b1 = 0;
end;
:: RSSPACE:def 6
theorem
for b1 being Element of the_set_of_RealSequences holds
b1 = Zeroseq
iff
for b2 being Element of NAT holds
(seq_id b1) . b2 = 0;
:: RSSPACE:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
seq_id b1 = b1;
:: RSSPACE:th 4
theorem
for b1, b2 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
b1 + b2 = (seq_id b1) + seq_id b2;
:: RSSPACE:th 5
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
b1 * b2 = b1 (#) seq_id b2;
:: RSSPACE:funcreg 1
registration
cluster RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) -> strict Abelian;
end;
:: RSSPACE:th 6
theorem
for b1, b2, b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
(b1 + b2) + b3 = b1 + (b2 + b3);
:: RSSPACE:th 7
theorem
for b1 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
b1 + 0. RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) = b1;
:: RSSPACE:th 8
theorem
for b1 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
ex b2 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) st
b1 + b2 = 0. RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#);
:: RSSPACE:th 9
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);
:: RSSPACE:th 10
theorem
for b1, b2 being Element of REAL
for b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
(b1 + b2) * b3 = (b1 * b3) + (b2 * b3);
:: RSSPACE:th 11
theorem
for b1, b2 being Element of REAL
for b3 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
(b1 * b2) * b3 = b1 * (b2 * b3);
:: RSSPACE:th 12
theorem
for b1 being Element of the carrier of RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#) holds
1 * b1 = b1;
:: RSSPACE:funcnot 7 => RSSPACE:func 7
definition
func Linear_Space_of_RealSequences -> RLSStruct equals
RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#);
end;
:: RSSPACE:def 7
theorem
Linear_Space_of_RealSequences = RLSStruct(#the_set_of_RealSequences,Zeroseq,l_add,l_mult#);
:: RSSPACE:funcreg 2
registration
cluster Linear_Space_of_RealSequences -> non empty strict;
end;
:: RSSPACE:funcreg 3
registration
cluster Linear_Space_of_RealSequences -> right_complementable Abelian add-associative right_zeroed RealLinearSpace-like;
end;
:: RSSPACE:funcnot 8 => RSSPACE:func 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
assume a2 is linearly-closed(a1) & a2 is not empty;
func Add_(A2,A1) -> Function-like quasi_total Relation of [:a2,a2:],a2 equals
(the addF of a1) || a2;
end;
:: RSSPACE:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-closed(b1) & b2 is not empty
holds Add_(b2,b1) = (the addF of b1) || b2;
:: RSSPACE:funcnot 9 => RSSPACE:func 9
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
assume a2 is linearly-closed(a1) & a2 is not empty;
func Mult_(A2,A1) -> Function-like quasi_total Relation of [:REAL,a2:],a2 equals
(the Mult of a1) | [:REAL,a2:];
end;
:: RSSPACE:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-closed(b1) & b2 is not empty
holds Mult_(b2,b1) = (the Mult of b1) | [:REAL,b2:];
:: RSSPACE:funcnot 10 => RSSPACE:func 10
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
assume a2 is linearly-closed(a1) & a2 is not empty;
func Zero_(A2,A1) -> Element of a2 equals
0. a1;
end;
:: RSSPACE:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-closed(b1) & b2 is not empty
holds Zero_(b2,b1) = 0. b1;
:: RSSPACE:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-closed(b1) & b2 is not empty
holds RLSStruct(#b2,Zero_(b2,b1),Add_(b2,b1),Mult_(b2,b1)#) is Subspace of b1;
:: RSSPACE:funcnot 11 => RSSPACE:func 11
definition
func the_set_of_l2RealSequences -> Element of bool the carrier of Linear_Space_of_RealSequences means
for b1 being set holds
b1 in it
iff
b1 in the_set_of_RealSequences & (seq_id b1) (#) seq_id b1 is summable;
end;
:: RSSPACE:def 11
theorem
for b1 being Element of bool the carrier of Linear_Space_of_RealSequences holds
b1 = the_set_of_l2RealSequences
iff
for b2 being set holds
b2 in b1
iff
b2 in the_set_of_RealSequences & (seq_id b2) (#) seq_id b2 is summable;
:: RSSPACE:funcreg 4
registration
cluster the_set_of_l2RealSequences -> non empty linearly-closed;
end;
:: RSSPACE:th 15
theorem
RLSStruct(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences)#) is Subspace of Linear_Space_of_RealSequences;
:: RSSPACE:th 16
theorem
RLSStruct(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences)#) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
:: RSSPACE:th 17
theorem
the carrier of Linear_Space_of_RealSequences = the_set_of_RealSequences &
(for b1 being set holds
b1 is Element of the carrier of Linear_Space_of_RealSequences
iff
b1 is Function-like quasi_total Relation of NAT,REAL) &
(for b1 being Element of the carrier of Linear_Space_of_RealSequences holds
b1 = seq_id b1) &
(for b1, b2 being Element of the carrier of Linear_Space_of_RealSequences holds
b1 + b2 = (seq_id b1) + seq_id b2) &
(for b1 being Element of REAL
for b2 being Element of the carrier of Linear_Space_of_RealSequences holds
b1 * b2 = b1 (#) seq_id b2);
:: RSSPACE:th 18
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL st
for b2, b3 being set
st b2 in the_set_of_l2RealSequences & b3 in the_set_of_l2RealSequences
holds b1 .(b2,b3) = Sum ((seq_id b2) (#) seq_id b3);
:: RSSPACE:funcnot 12 => RSSPACE:func 12
definition
func l_scalar -> Function-like quasi_total Relation of [:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL means
for b1, b2 being set
st b1 in the_set_of_l2RealSequences & b2 in the_set_of_l2RealSequences
holds it .(b1,b2) = Sum ((seq_id b1) (#) seq_id b2);
end;
:: RSSPACE:def 12
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL holds
b1 = l_scalar
iff
for b2, b3 being set
st b2 in the_set_of_l2RealSequences & b3 in the_set_of_l2RealSequences
holds b1 .(b2,b3) = Sum ((seq_id b2) (#) seq_id b3);
:: RSSPACE:funcreg 5
registration
cluster UNITSTR(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),l_scalar#) -> non empty strict;
end;
:: RSSPACE:funcnot 13 => RSSPACE:func 13
definition
func l2_Space -> non empty UNITSTR equals
UNITSTR(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),l_scalar#);
end;
:: RSSPACE:def 13
theorem
l2_Space = UNITSTR(#the_set_of_l2RealSequences,Zero_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l2RealSequences,Linear_Space_of_RealSequences),l_scalar#);
:: RSSPACE:th 19
theorem
for b1 being RLSStruct
st RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
holds b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
:: RSSPACE:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st for b2 being Element of NAT holds
b1 . b2 = 0
holds b1 is summable & Sum b1 = 0;
:: RSSPACE:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st (for b2 being Element of NAT holds
0 <= b1 . b2) &
b1 is summable &
Sum b1 = 0
for b2 being Element of NAT holds
b1 . b2 = 0;
:: RSSPACE:funcreg 6
registration
cluster l2_Space -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like;
end;