:: Introduction to Banach and Hilbert spaces - Part III
:: by Jan Popio{\l}ek
::
:: Copyright (c) 1991-2017 Association of Mizar Users

deffunc H1( RealUnitarySpace) -> Element of the U1 of \$1 = 0. \$1;

registration
let X be RealUnitarySpace;
cluster V1() V4( NAT ) V5( the U1 of X) V6() constant V11() V14( NAT ) V18( NAT , the U1 of X) for Element of K19(K20(NAT, the U1 of X));
existence
ex b1 being sequence of X st b1 is constant
proof end;
end;

definition
let X be RealUnitarySpace;
let seq be sequence of X;
attr seq is Cauchy means :Def1: :: BHSP_3:def 1
for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
dist ((seq . n),(seq . m)) < r;
end;

:: deftheorem Def1 defines Cauchy BHSP_3:def 1 :
for X being RealUnitarySpace
for seq being sequence of X holds
( seq is Cauchy 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
dist ((seq . n),(seq . m)) < r );

registration
let X be RealUnitarySpace;
cluster V6() constant V18( NAT , the U1 of X) -> Cauchy for Element of K19(K20(NAT, the U1 of X));
coherence
for b1 being sequence of X st b1 is constant holds
b1 is Cauchy
proof end;
end;

theorem :: BHSP_3:1
canceled;

::\$CT
theorem :: BHSP_3:2
for X being RealUnitarySpace
for seq being sequence of X holds
( seq is Cauchy 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;

registration
let X be RealUnitarySpace;
let seq1, seq2 be Cauchy sequence of X;
cluster seq1 + seq2 -> Cauchy ;
coherence
seq1 + seq2 is Cauchy
proof end;
end;

registration
let X be RealUnitarySpace;
let seq1, seq2 be Cauchy sequence of X;
cluster seq1 - seq2 -> Cauchy ;
coherence
seq1 - seq2 is Cauchy
proof end;
end;

registration
let X be RealUnitarySpace;
let a be Real;
let seq be Cauchy sequence of X;
cluster a * seq -> Cauchy ;
coherence
a * seq is Cauchy
proof end;
end;

registration
let X be RealUnitarySpace;
let seq be Cauchy sequence of X;
cluster - seq -> Cauchy ;
coherence
- seq is Cauchy
proof end;
end;

registration
let X be RealUnitarySpace;
let x be Point of X;
let seq be Cauchy sequence of X;
cluster seq + x -> Cauchy ;
coherence
seq + x is Cauchy
proof end;
end;

registration
let X be RealUnitarySpace;
let x be Point of X;
let seq be Cauchy sequence of X;
cluster seq - x -> Cauchy ;
coherence
seq - x is Cauchy
proof end;
end;

registration
let X be RealUnitarySpace;
cluster V6() V18( NAT , the U1 of X) convergent -> Cauchy for Element of K19(K20(NAT, the U1 of X));
coherence
for b1 being sequence of X st b1 is convergent holds
b1 is Cauchy
proof end;
end;

definition
let X be RealUnitarySpace;
let seq1, seq2 be sequence of X;
pred seq1 is_compared_to seq2 means :: BHSP_3:def 2
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r;
reflexivity
for seq1 being sequence of X
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq1 . n)) < r
proof end;
symmetry
for seq1, seq2 being sequence of X st ( for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r ) holds
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq2 . n),(seq1 . n)) < r
proof end;
end;

:: deftheorem defines is_compared_to BHSP_3:def 2 :
for X being RealUnitarySpace
for seq1, seq2 being sequence of X holds
( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r );

theorem :: BHSP_3:3
canceled;

theorem :: BHSP_3:4
canceled;

theorem :: BHSP_3:5
canceled;

theorem :: BHSP_3:6
canceled;

theorem :: BHSP_3:7
canceled;

theorem :: BHSP_3:8
canceled;

theorem :: BHSP_3:9
canceled;

theorem :: BHSP_3:10
canceled;

theorem :: BHSP_3:11
canceled;

::\$CT 9
theorem :: BHSP_3:12
for X being RealUnitarySpace
for seq1, seq2, seq3 being sequence of X st seq1 is_compared_to seq2 & seq2 is_compared_to seq3 holds
seq1 is_compared_to seq3
proof end;

theorem :: BHSP_3:13
for X being RealUnitarySpace
for seq1, seq2 being sequence of X holds
( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r )
proof end;

theorem :: BHSP_3:14
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st ex k being Nat st
for n being Nat st n >= k holds
seq1 . n = seq2 . n holds
seq1 is_compared_to seq2
proof end;

theorem :: BHSP_3:15
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is Cauchy & seq1 is_compared_to seq2 holds
seq2 is Cauchy
proof end;

theorem :: BHSP_3:16
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent & seq1 is_compared_to seq2 holds
seq2 is convergent
proof end;

theorem :: BHSP_3:17
for X being RealUnitarySpace
for g being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 holds
( seq2 is convergent & lim seq2 = g )
proof end;

definition
let X be RealUnitarySpace;
let seq be sequence of X;
attr seq is bounded means :Def3: :: BHSP_3:def 3
ex M being Real st
( M > 0 & ( for n being Nat holds ||.(seq . n).|| <= M ) );
end;

:: deftheorem Def3 defines bounded BHSP_3:def 3 :
for X being RealUnitarySpace
for seq being sequence of X holds
( seq is bounded iff ex M being Real st
( M > 0 & ( for n being Nat holds ||.(seq . n).|| <= M ) ) );

registration
let X be RealUnitarySpace;
cluster V6() constant V18( NAT , the U1 of X) -> bounded for Element of K19(K20(NAT, the U1 of X));
coherence
for b1 being sequence of X st b1 is constant holds
b1 is bounded
proof end;
end;

registration
let X be RealUnitarySpace;
let seq1, seq2 be bounded sequence of X;
cluster seq1 + seq2 -> bounded ;
coherence
seq1 + seq2 is bounded
proof end;
end;

registration
let X be RealUnitarySpace;
let seq be bounded sequence of X;
cluster - seq -> bounded ;
coherence
- seq is bounded
proof end;
end;

registration
let X be RealUnitarySpace;
let seq1, seq2 be bounded sequence of X;
cluster seq1 - seq2 -> bounded ;
coherence
seq1 - seq2 is bounded
proof end;
end;

registration
let X be RealUnitarySpace;
let a be Real;
let seq be bounded sequence of X;
cluster a * seq -> bounded ;
coherence
a * seq is bounded
proof end;
end;

theorem :: BHSP_3:18
canceled;

theorem :: BHSP_3:19
canceled;

theorem :: BHSP_3:20
canceled;

theorem :: BHSP_3:21
canceled;

theorem :: BHSP_3:22
canceled;

::\$CT 5
theorem Th8: :: BHSP_3:23
for X being RealUnitarySpace
for seq being sequence of X
for m being Nat ex M being Real st
( M > 0 & ( for n being Nat st n <= m holds
||.(seq . n).|| < M ) )
proof end;

registration
let X be RealUnitarySpace;
cluster V6() V18( NAT , the U1 of X) convergent -> bounded for Element of K19(K20(NAT, the U1 of X));
coherence
for b1 being sequence of X st b1 is convergent holds
b1 is bounded
proof end;
end;

theorem :: BHSP_3:24
canceled;

::\$CT
theorem :: BHSP_3:25
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is bounded & seq1 is_compared_to seq2 holds
seq2 is bounded
proof end;

registration
let X be RealUnitarySpace;
let seq be bounded sequence of X;
cluster -> bounded for subsequence of seq;
coherence
for b1 being subsequence of seq holds b1 is bounded
proof end;
end;

registration
let X be RealUnitarySpace;
let seq be convergent sequence of X;
cluster -> convergent for subsequence of seq;
coherence
for b1 being subsequence of seq holds b1 is convergent
proof end;
end;

theorem :: BHSP_3:26
canceled;

theorem :: BHSP_3:27
canceled;

::\$CT 2
theorem Th10: :: BHSP_3:28
for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq1 is subsequence of seq & seq is convergent holds
lim seq1 = lim seq
proof end;

registration
let X be RealUnitarySpace;
let seq be Cauchy sequence of X;
cluster -> Cauchy for subsequence of seq;
coherence
for b1 being subsequence of seq holds b1 is Cauchy
proof end;
end;

theorem :: BHSP_3:29
canceled;

::\$CT
theorem :: BHSP_3:30
for X being RealUnitarySpace
for seq being sequence of X
for k, m being Nat holds (seq ^\ k) ^\ m = (seq ^\ m) ^\ k
proof end;

theorem :: BHSP_3:31
for X being RealUnitarySpace
for seq being sequence of X
for k, m being Nat holds (seq ^\ k) ^\ m = seq ^\ (k + m)
proof end;

theorem Th13: :: BHSP_3:32
for X being RealUnitarySpace
for seq1, seq2 being sequence of X
for k being Nat holds (seq1 + seq2) ^\ k = (seq1 ^\ k) + (seq2 ^\ k)
proof end;

theorem Th14: :: BHSP_3:33
for X being RealUnitarySpace
for seq being sequence of X
for k being Nat holds (- seq) ^\ k = - (seq ^\ k)
proof end;

theorem :: BHSP_3:34
for X being RealUnitarySpace
for seq1, seq2 being sequence of X
for k being Nat holds (seq1 - seq2) ^\ k = (seq1 ^\ k) - (seq2 ^\ k)
proof end;

theorem :: BHSP_3:35
for X being RealUnitarySpace
for a being Real
for seq being sequence of X
for k being Nat holds (a * seq) ^\ k = a * (seq ^\ k)
proof end;

theorem :: BHSP_3:36
for X being RealUnitarySpace
for seq being sequence of X
for k being Nat st seq is convergent holds
( seq ^\ k is convergent & lim (seq ^\ k) = lim seq ) by Th10;

theorem :: BHSP_3:37
for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq is convergent & ex k being Nat st seq = seq1 ^\ k holds
seq1 is convergent
proof end;

theorem :: BHSP_3:38
for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq is Cauchy & ex k being Nat st seq = seq1 ^\ k holds
seq1 is Cauchy
proof end;

theorem :: BHSP_3:39
canceled;

::\$CT
theorem :: BHSP_3:40
for X being RealUnitarySpace
for seq1, seq2 being sequence of X
for k being Nat st seq1 is_compared_to seq2 holds
seq1 ^\ k is_compared_to seq2 ^\ k
proof end;

definition
let X be RealUnitarySpace;
attr X is complete means :: BHSP_3:def 4
for seq being sequence of X st seq is Cauchy holds
seq is convergent ;
end;

:: deftheorem defines complete BHSP_3:def 4 :
for X being RealUnitarySpace holds
( X is complete iff for seq being sequence of X st seq is Cauchy holds
seq is convergent );

theorem :: BHSP_3:41
canceled;

theorem :: BHSP_3:42
canceled;

::\$CT 2
theorem :: BHSP_3:43
for X being RealUnitarySpace
for seq being sequence of X st X is complete & seq is Cauchy holds
seq is bounded
proof end;