:: Introduction to Banach and Hilbert spaces - Part III
:: by Jan Popio{\l}ek
::
:: Received July 19, 1991
:: Copyright (c) 1991-2011 Association of Mizar Users

begin

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

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 Element of NAT st
for n, m being Element of 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 Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
dist ((seq . n),(seq . m)) < r );

theorem :: BHSP_3:1

for X being RealUnitarySpace
for seq being sequence of X st seq is V8() holds
seq is Cauchy

proof end;

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 Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
||.((seq . n) - (seq . m)).|| < r )

proof end;

theorem :: BHSP_3:3

for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds
seq1 + seq2 is Cauchy

proof end;

theorem :: BHSP_3:4

for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds
seq1 - seq2 is Cauchy

proof end;

theorem Th5: :: BHSP_3:5

for X being RealUnitarySpace
for a being Real
for seq being sequence of X st seq is Cauchy holds
a * seq is Cauchy

proof end;

theorem :: BHSP_3:6

for X being RealUnitarySpace
for seq being sequence of X st seq is Cauchy holds
- seq is Cauchy

proof end;

theorem Th7: :: BHSP_3:7

for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is Cauchy holds
seq + x is Cauchy

proof end;

theorem :: BHSP_3:8

for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is Cauchy holds
seq - x is Cauchy

proof end;

theorem :: BHSP_3:9

for X being RealUnitarySpace
for seq being sequence of X st seq is convergent holds
seq is Cauchy

proof end;

definition

let X be RealUnitarySpace;
let seq1, seq2 be sequence of X;
pred seq1 is_compared_to seq2 means :Def2: :: BHSP_3:def 2
for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r;

end;

:: deftheorem Def2 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 Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r );

theorem Th10: :: BHSP_3:10

for X being RealUnitarySpace
for seq being sequence of X holds seq is_compared_to seq

proof end;

theorem Th11: :: BHSP_3:11

for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is_compared_to seq2 holds
seq2 is_compared_to seq1

proof end;

definition

let X be RealUnitarySpace;
let seq1, seq2 be sequence of X;
:: original: is_compared_to
redefine pred seq1 is_compared_to seq2;
reflexivity
for seq1 being sequence of X holds seq1 is_compared_to seq1 by Th10;
symmetry
for seq1, seq2 being sequence of X st seq1 is_compared_to seq2 holds
seq2 is_compared_to seq1 by Th11;

end;

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 Element of NAT st
for n being Element of 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 Element of NAT st
for n being Element of 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 Element of 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 Element of NAT holds ||.(seq . n).|| <= M ) ) );

theorem Th18: :: BHSP_3:18

for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds
seq1 + seq2 is bounded

proof end;

theorem Th19: :: BHSP_3:19

for X being RealUnitarySpace
for seq being sequence of X st seq is bounded holds
- seq is bounded

proof end;

theorem :: BHSP_3:20

for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds
seq1 - seq2 is bounded

proof end;

theorem :: BHSP_3:21

for X being RealUnitarySpace
for a being Real
for seq being sequence of X st seq is bounded holds
a * seq is bounded

proof end;

theorem :: BHSP_3:22

for X being RealUnitarySpace
for seq being sequence of X st seq is V8() holds
seq is bounded

proof end;

theorem Th23: :: BHSP_3:23

for X being RealUnitarySpace
for seq being sequence of X
for m being Element of NAT ex M being Real st
( M > 0 & ( for n being Element of NAT st n <= m holds
||.(seq . n).|| < M ) )

proof end;

theorem Th24: :: BHSP_3:24

for X being RealUnitarySpace
for seq being sequence of X st seq is convergent holds
seq is bounded

proof end;

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;

theorem :: BHSP_3:26

canceled;

theorem :: BHSP_3:27

canceled;

theorem :: BHSP_3:28

canceled;

theorem :: BHSP_3:29

canceled;

theorem :: BHSP_3:30

canceled;

theorem Th31: :: BHSP_3:31

for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq is bounded & seq1 is subsequence of seq holds
seq1 is bounded

proof end;

theorem Th32: :: BHSP_3:32

for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq is convergent & seq1 is subsequence of seq holds
seq1 is convergent

proof end;

theorem Th33: :: BHSP_3:33

for X being RealUnitarySpace
for seq1, seq being sequence of X st seq1 is subsequence of seq & seq is convergent holds
lim seq1 = lim seq

proof end;

theorem Th34: :: BHSP_3:34

for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq is Cauchy & seq1 is subsequence of seq holds
seq1 is Cauchy

proof end;

theorem :: BHSP_3:35

canceled;

theorem :: BHSP_3:36

for X being RealUnitarySpace
for seq being sequence of X
for k, m being Element of NAT holds (seq ^\ k) ^\ m = (seq ^\ m) ^\ k

proof end;

theorem :: BHSP_3:37

for X being RealUnitarySpace
for seq being sequence of X
for k, m being Element of NAT holds (seq ^\ k) ^\ m = seq ^\ (k + m)

proof end;

theorem Th38: :: BHSP_3:38

for X being RealUnitarySpace
for seq1, seq2 being sequence of X
for k being Element of NAT holds (seq1 + seq2) ^\ k = (seq1 ^\ k) + (seq2 ^\ k)

proof end;

theorem Th39: :: BHSP_3:39

for X being RealUnitarySpace
for seq being sequence of X
for k being Element of NAT holds (- seq) ^\ k = - (seq ^\ k)

proof end;

theorem :: BHSP_3:40

for X being RealUnitarySpace
for seq1, seq2 being sequence of X
for k being Element of NAT holds (seq1 - seq2) ^\ k = (seq1 ^\ k) - (seq2 ^\ k)

proof end;

theorem :: BHSP_3:41

for X being RealUnitarySpace
for a being Real
for seq being sequence of X
for k being Element of NAT holds (a * seq) ^\ k = a * (seq ^\ k)

proof end;

theorem :: BHSP_3:42

canceled;

theorem :: BHSP_3:43

canceled;

theorem :: BHSP_3:44

for X being RealUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is convergent holds
( seq ^\ k is convergent & lim (seq ^\ k) = lim seq ) by Th32, Th33;

theorem :: BHSP_3:45

canceled;

theorem :: BHSP_3:46

for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq is convergent & ex k being Element of NAT st seq = seq1 ^\ k holds
seq1 is convergent

proof end;

theorem :: BHSP_3:47

for X being RealUnitarySpace
for seq, seq1 being sequence of X st seq is Cauchy & ex k being Element of NAT st seq = seq1 ^\ k holds
seq1 is Cauchy

proof end;

theorem :: BHSP_3:48

for X being RealUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is Cauchy holds
seq ^\ k is Cauchy by Th34;

theorem :: BHSP_3:49

for X being RealUnitarySpace
for seq1, seq2 being sequence of X
for k being Element of NAT st seq1 is_compared_to seq2 holds
seq1 ^\ k is_compared_to seq2 ^\ k

proof end;

theorem :: BHSP_3:50

for X being RealUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is bounded holds
seq ^\ k is bounded by Th31;

theorem :: BHSP_3:51

for X being RealUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is V8() holds
seq ^\ k is V8() ;

definition

canceled;
canceled;
let X be RealUnitarySpace;
attr X is complete means :Def6: :: BHSP_3:def 6
for seq being sequence of X st seq is Cauchy holds
seq is convergent ;

end;

:: deftheorem BHSP_3:def 4 :
canceled;

:: deftheorem BHSP_3:def 5 :
canceled;

:: deftheorem Def6 defines complete BHSP_3:def 6 :
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_3:52

canceled;

theorem :: BHSP_3:53

for X being RealUnitarySpace
for seq being sequence of X st X is complete & seq is Cauchy holds
seq is bounded

proof end;