RSSPACE3: Banach Space of Absolute Summable Real Sequences

:: Banach Space of Absolute Summable Real Sequences
:: by Yasumasa Suzuki, Noboru Endou and Yasunari Shidama
::
:: Received August 8, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

definition

func the_set_of_l1RealSequences -> Subset of Linear_Space_of_RealSequences means :Def1: :: RSSPACE3:def 1
for x being object holds
( x in it iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines the_set_of_l1RealSequences RSSPACE3:def 1 :

theorem Th1: :: RSSPACE3:1

for c being Real
for seq being Real_Sequence st seq is convergent holds
for rseq being Real_Sequence st ( for i being Nat holds rseq . i = |.((seq . i) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| )

proof end;

registration

cluster the_set_of_l1RealSequences -> non empty ;

coherence

proof end;

end;

registration

cluster the_set_of_l1RealSequences -> linearly-closed ;

coherence

proof end;

end;

Lm1: RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is Subspace of Linear_Space_of_RealSequences
by RSSPACE:11;

registration

cluster RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

coherence by RSSPACE:11;

end;

Lm2: RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is RealLinearSpace
;

definition

func l_norm -> Function of the_set_of_l1RealSequences,REAL means :Def2: :: RSSPACE3:def 2
for x being object st x in the_set_of_l1RealSequences holds
it . x = Sum (abs (seq_id x));

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines l_norm RSSPACE3:def 2 :

registration

let X be non empty set ;
let Z be Element of X;
let A be BinOp of X;
let M be Function of [:REAL,X:],X;
let N be Function of X,REAL;

cluster NORMSTR(# X,Z,A,M,N #) -> non empty ;

coherence ;

end;

theorem :: RSSPACE3:2

for l being NORMSTR st RLSStruct(# the carrier of l, the ZeroF of l, the addF of l, the Mult of l #) is RealLinearSpace holds
l is RealLinearSpace by RSSPACE:15;

theorem Th3: :: RSSPACE3:3

for rseq being Real_Sequence st ( for n being Nat holds rseq . n = 0 ) holds
( rseq is absolutely_summable & Sum (abs rseq) = 0 )

proof end;

theorem Th4: :: RSSPACE3:4

for rseq being Real_Sequence st rseq is absolutely_summable & Sum (abs rseq) = 0 holds
for n being Nat holds rseq . n = 0

proof end;

theorem Th5: :: RSSPACE3:5

NORMSTR(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),l_norm #) is RealLinearSpace by Lm2, RSSPACE:15;

definition end;

:: deftheorem defines l1_Space RSSPACE3:def 3 :

theorem Th6: :: RSSPACE3:6

( the carrier of l1_Space = the_set_of_l1RealSequences & ( for x being set holds
( x is VECTOR of l1_Space iff ( x is Real_Sequence & seq_id x is absolutely_summable ) ) ) & 0. l1_Space = Zeroseq & ( for u being VECTOR of l1_Space holds u = seq_id u ) & ( for u, v being VECTOR of l1_Space holds u + v = (seq_id u) + (seq_id v) ) & ( for r being Real
for u being VECTOR of l1_Space holds r * u = r (#) (seq_id u) ) & ( for u being VECTOR of l1_Space holds
( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of l1_Space holds u - v = (seq_id u) - (seq_id v) ) & ( for v being VECTOR of l1_Space holds seq_id v is absolutely_summable ) & ( for v being VECTOR of l1_Space holds ||.v.|| = Sum (abs (seq_id v)) ) )

proof end;

theorem Th7: :: RSSPACE3:7

for x, y being Point of l1_Space
for a being Real holds
( ( ||.x.|| = 0 implies x = 0. l1_Space ) & ( x = 0. l1_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = |.a.| * ||.x.|| )

proof end;

registration end;

Lm3: for c being Real
for seq, seq1 being Real_Sequence st seq is convergent & seq1 is convergent holds
for rseq being Real_Sequence st ( for i being Nat holds rseq . i = |.((seq . i) - c).| + (seq1 . i) ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| + (lim seq1) )

proof end;

definition

let X be non empty NORMSTR ;
let x, y be Point of X;

func dist (x,y) -> Real equals :: RSSPACE3:def 4
||.(x - y).||;

coherence ;

end;

:: deftheorem defines dist RSSPACE3:def 4 :

definition

let NRM be non empty NORMSTR ;
let seqt be sequence of NRM;

attr seqt is CCauchy means :: RSSPACE3:def 5
for r1 being Real st r1 > 0 holds
ex k1 being Nat st
for n1, m1 being Nat st n1 >= k1 & m1 >= k1 holds
dist ((seqt . n1),(seqt . m1)) < r1;

end;

:: deftheorem defines CCauchy RSSPACE3:def 5 :

notation

let NRM be non empty NORMSTR ;
let seqt be sequence of NRM;
synonym Cauchy_sequence_by_Norm seqt for CCauchy ;

end;

theorem Th8: :: RSSPACE3:8

for NRM being RealNormSpace
for seq being sequence of NRM holds
( seq is Cauchy_sequence_by_Norm iff for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((seq . n) - (seq . m)).|| < r )

proof end;

theorem :: RSSPACE3:9

for vseq being sequence of l1_Space st vseq is Cauchy_sequence_by_Norm holds
vseq is convergent

proof end;