:: Introduction to Banach and Hilbert spaces - Part II
:: by Jan Popio{\l}ek
::
:: Received July 19, 1991
:: Copyright (c) 1991-2012 Association of Mizar Users


begin

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

definition
let X be RealUnitarySpace;
let seq be sequence of X;
attrseq is convergent  means :Def1: :: BHSP_2:def 1
 ex g being Point of X st
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 ((seq . n),g) < r;
end;

:: deftheorem Def1 defines convergent BHSP_2:def_1_:_
 for X being RealUnitarySpace
for seq being sequence of X holds
( seq is convergent iff ex g being Point of X st
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 ((seq . n),g) < r );

theorem Th1: :: BHSP_2:1
for X being RealUnitarySpace
for seq being sequence of X st seq is constant holds
seq is convergent
proofend;

theorem Th2: :: BHSP_2:2
for X being RealUnitarySpace
for seq, seq9 being sequence of X st seq is convergent &amp; ex k being Element of NAT st
for n being Element of NAT st k <= n holds
seq9 . n = seq . n holds
seq9 is convergent
proofend;

theorem Th3: :: BHSP_2:3
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent &amp; seq2 is convergent holds
seq1 + seq2 is convergent
proofend;

theorem Th4: :: BHSP_2:4
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent &amp; seq2 is convergent holds
seq1 - seq2 is convergent
proofend;

theorem Th5: :: BHSP_2:5
for X being RealUnitarySpace
for a being Real
for seq being sequence of X st seq is convergent holds
a * seq is convergent
proofend;

theorem Th6: :: BHSP_2:6
for X being RealUnitarySpace
for seq being sequence of X st seq is convergent holds
- seq is convergent
proofend;

theorem Th7: :: BHSP_2:7
for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
seq + x is convergent
proofend;

theorem Th8: :: BHSP_2:8
for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
seq - x is convergent
proofend;

theorem Th9: :: BHSP_2:9
for X being RealUnitarySpace
for seq being sequence of X holds
( seq is convergent iff ex g being Point of X st
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
||.((seq . n) - g).|| < r )
proofend;

definition
let X be RealUnitarySpace;
let seq be sequence of X;
assume A1: seq is convergent ;
func lim seq ->  Point of X means :Def2: :: BHSP_2: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 ((seq . n),it) < r;
existence
 ex b1 being Point of X st
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 ((seq . n),b1) < r by A1, Def1;
uniqueness
 for b1, b2 being Point of X st ( 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 ((seq . n),b1) < r ) &amp; ( 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 ((seq . n),b2) < r ) holds
b1 = b2
proofend;
end;

:: deftheorem Def2 defines lim BHSP_2:def_2_:_
 for X being RealUnitarySpace
for seq being sequence of X st seq is convergent holds
for b3 being Point of X holds
( b3 = lim seq 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 ((seq . n),b3) < r );

theorem Th10: :: BHSP_2:10
for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is constant &amp; x in rng seq holds
lim seq = x
proofend;

theorem :: BHSP_2:11
for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is constant &amp; ex n being Element of NAT st seq . n = x holds
lim seq = x
proofend;

theorem :: BHSP_2:12
for X being RealUnitarySpace
for seq, seq9 being sequence of X st seq is convergent &amp; ex k being Element of NAT st
for n being Element of NAT st n >= k holds
seq9 . n = seq . n holds
lim seq = lim seq9
proofend;

theorem Th13: :: BHSP_2:13
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent &amp; seq2 is convergent holds
lim (seq1 + seq2) = (lim seq1) + (lim seq2)
proofend;

theorem Th14: :: BHSP_2:14
for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent &amp; seq2 is convergent holds
lim (seq1 - seq2) = (lim seq1) - (lim seq2)
proofend;

theorem Th15: :: BHSP_2:15
for X being RealUnitarySpace
for a being Real
for seq being sequence of X st seq is convergent holds
lim (a * seq) = a * (lim seq)
proofend;

theorem Th16: :: BHSP_2:16
for X being RealUnitarySpace
for seq being sequence of X st seq is convergent holds
lim (- seq) = - (lim seq)
proofend;

theorem Th17: :: BHSP_2:17
for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
lim (seq + x) = (lim seq) + x
proofend;

theorem :: BHSP_2:18
for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
lim (seq - x) = (lim seq) - x
proofend;

theorem Th19: :: BHSP_2:19
for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent holds
( lim seq = g 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
||.((seq . n) - g).|| < r )
proofend;

definition
let X be RealUnitarySpace;
let seq be sequence of X;
func||.seq.|| ->  Real_Sequence means :Def3: :: BHSP_2:def 3
 for n being Element of NAT holds it . n = ||.(seq . n).||;
existence
 ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = ||.(seq . n).||
proofend;
uniqueness
 for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = ||.(seq . n).|| ) &amp; ( for n being Element of NAT holds b2 . n = ||.(seq . n).|| ) holds
b1 = b2
proofend;
end;

:: deftheorem Def3 defines ||. BHSP_2:def_3_:_
 for X being RealUnitarySpace
for seq being sequence of X
for b3 being Real_Sequence holds
( b3 = ||.seq.|| iff for n being Element of NAT holds b3 . n = ||.(seq . n).|| );

theorem Th20: :: BHSP_2:20
for X being RealUnitarySpace
for seq being sequence of X st seq is convergent holds
||.seq.|| is convergent
proofend;

theorem Th21: :: BHSP_2:21
for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.seq.|| is convergent &amp; lim ||.seq.|| = ||.g.|| )
proofend;

theorem Th22: :: BHSP_2:22
for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.(seq - g).|| is convergent &amp; lim ||.(seq - g).|| = 0 )
proofend;

definition
let X be RealUnitarySpace;
let seq be sequence of X;
let x be Point of X;
func dist (seq,x) ->  Real_Sequence means :Def4: :: BHSP_2:def 4
 for n being Element of NAT holds it . n = dist ((seq . n),x);
existence
 ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = dist ((seq . n),x)
proofend;
uniqueness
 for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = dist ((seq . n),x) ) &amp; ( for n being Element of NAT holds b2 . n = dist ((seq . n),x) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def4 defines dist BHSP_2:def_4_:_
 for X being RealUnitarySpace
for seq being sequence of X
for x being Point of X
for b4 being Real_Sequence holds
( b4 = dist (seq,x) iff for n being Element of NAT holds b4 . n = dist ((seq . n),x) );

theorem Th23: :: BHSP_2:23
for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
dist (seq,g) is convergent
proofend;

theorem Th24: :: BHSP_2:24
for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( dist (seq,g) is convergent &amp; lim (dist (seq,g)) = 0 )
proofend;

theorem :: BHSP_2:25
for X being RealUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent &amp; lim seq1 = g1 &amp; seq2 is convergent &amp; lim seq2 = g2 holds
( ||.(seq1 + seq2).|| is convergent &amp; lim ||.(seq1 + seq2).|| = ||.(g1 + g2).|| )
proofend;

theorem :: BHSP_2:26
for X being RealUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent &amp; lim seq1 = g1 &amp; seq2 is convergent &amp; lim seq2 = g2 holds
( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent &amp; lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )
proofend;

theorem :: BHSP_2:27
for X being RealUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent &amp; lim seq1 = g1 &amp; seq2 is convergent &amp; lim seq2 = g2 holds
( ||.(seq1 - seq2).|| is convergent &amp; lim ||.(seq1 - seq2).|| = ||.(g1 - g2).|| )
proofend;

theorem :: BHSP_2:28
for X being RealUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent &amp; lim seq1 = g1 &amp; seq2 is convergent &amp; lim seq2 = g2 holds
( ||.((seq1 - seq2) - (g1 - g2)).|| is convergent &amp; lim ||.((seq1 - seq2) - (g1 - g2)).|| = 0 )
proofend;

theorem :: BHSP_2:29
for X being RealUnitarySpace
for g being Point of X
for a being Real
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.(a * seq).|| is convergent &amp; lim ||.(a * seq).|| = ||.(a * g).|| )
proofend;

theorem :: BHSP_2:30
for X being RealUnitarySpace
for g being Point of X
for a being Real
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.((a * seq) - (a * g)).|| is convergent &amp; lim ||.((a * seq) - (a * g)).|| = 0 )
proofend;

theorem :: BHSP_2:31
for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.(- seq).|| is convergent &amp; lim ||.(- seq).|| = ||.(- g).|| )
proofend;

theorem :: BHSP_2:32
for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.((- seq) - (- g)).|| is convergent &amp; lim ||.((- seq) - (- g)).|| = 0 )
proofend;

Lm1:  for X being RealUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.(seq + x).|| is convergent &amp; lim ||.(seq + x).|| = ||.(g + x).|| )
proofend;

theorem Th33: :: BHSP_2:33
for X being RealUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.((seq + x) - (g + x)).|| is convergent &amp; lim ||.((seq + x) - (g + x)).|| = 0 )
proofend;

theorem :: BHSP_2:34
for X being RealUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.(seq - x).|| is convergent &amp; lim ||.(seq - x).|| = ||.(g - x).|| )
proofend;

theorem :: BHSP_2:35
for X being RealUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( ||.((seq - x) - (g - x)).|| is convergent &amp; lim ||.((seq - x) - (g - x)).|| = 0 )
proofend;

theorem :: BHSP_2:36
for X being RealUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent &amp; lim seq1 = g1 &amp; seq2 is convergent &amp; lim seq2 = g2 holds
( dist ((seq1 + seq2),(g1 + g2)) is convergent &amp; lim (dist ((seq1 + seq2),(g1 + g2))) = 0 )
proofend;

theorem :: BHSP_2:37
for X being RealUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent &amp; lim seq1 = g1 &amp; seq2 is convergent &amp; lim seq2 = g2 holds
( dist ((seq1 - seq2),(g1 - g2)) is convergent &amp; lim (dist ((seq1 - seq2),(g1 - g2))) = 0 )
proofend;

theorem :: BHSP_2:38
for X being RealUnitarySpace
for g being Point of X
for a being Real
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( dist ((a * seq),(a * g)) is convergent &amp; lim (dist ((a * seq),(a * g))) = 0 )
proofend;

theorem :: BHSP_2:39
for X being RealUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent &amp; lim seq = g holds
( dist ((seq + x),(g + x)) is convergent &amp; lim (dist ((seq + x),(g + x))) = 0 )
proofend;

definition
let X be RealUnitarySpace;
let x be Point of X;
let r be Real;
func Ball (x,r) ->  Subset of X equals :: BHSP_2:def 5
 {  y where y is Point of X : ||.(x - y).|| < r  }  ;
coherence
  {  y where y is Point of X : ||.(x - y).|| < r  }  is Subset of X
proofend;
func cl_Ball (x,r) ->  Subset of X equals :: BHSP_2:def 6
 {  y where y is Point of X : ||.(x - y).|| <= r  }  ;
coherence
  {  y where y is Point of X : ||.(x - y).|| <= r  }  is Subset of X
proofend;
func Sphere (x,r) ->  Subset of X equals :: BHSP_2:def 7
 {  y where y is Point of X : ||.(x - y).|| = r  }  ;
coherence
  {  y where y is Point of X : ||.(x - y).|| = r  }  is Subset of X
proofend;
end;

:: deftheorem  defines Ball BHSP_2:def_5_:_
 for X being RealUnitarySpace
for x being Point of X
for r being Real holds Ball (x,r) =  {  y where y is Point of X : ||.(x - y).|| < r  }  ;

:: deftheorem  defines cl_Ball BHSP_2:def_6_:_
 for X being RealUnitarySpace
for x being Point of X
for r being Real holds cl_Ball (x,r) =  {  y where y is Point of X : ||.(x - y).|| <= r  }  ;

:: deftheorem  defines Sphere BHSP_2:def_7_:_
 for X being RealUnitarySpace
for x being Point of X
for r being Real holds Sphere (x,r) =  {  y where y is Point of X : ||.(x - y).|| = r  }  ;

theorem Th40: :: BHSP_2:40
for X being RealUnitarySpace
for z, x being Point of X
for r being Real holds
( z in Ball (x,r) iff ||.(x - z).|| < r )
proofend;

theorem Th41: :: BHSP_2:41
for X being RealUnitarySpace
for z, x being Point of X
for r being Real holds
( z in Ball (x,r) iff dist (x,z) < r )
proofend;

theorem :: BHSP_2:42
for X being RealUnitarySpace
for x being Point of X
for r being Real st r > 0 holds
x in Ball (x,r)
proofend;

theorem :: BHSP_2:43
for X being RealUnitarySpace
for y, x, z being Point of X
for r being Real st y in Ball (x,r) &amp; z in Ball (x,r) holds
dist (y,z) < 2 * r
proofend;

theorem :: BHSP_2:44
for X being RealUnitarySpace
for y, x, z being Point of X
for r being Real st y in Ball (x,r) holds
y - z in Ball ((x - z),r)
proofend;

theorem :: BHSP_2:45
for X being RealUnitarySpace
for y, x being Point of X
for r being Real st y in Ball (x,r) holds
y - x in Ball ((0. X),r)
proofend;

theorem :: BHSP_2:46
for X being RealUnitarySpace
for y, x being Point of X
for r, q being Real st y in Ball (x,r) &amp; r <= q holds
y in Ball (x,q)
proofend;

theorem Th47: :: BHSP_2:47
for X being RealUnitarySpace
for z, x being Point of X
for r being Real holds
( z in cl_Ball (x,r) iff ||.(x - z).|| <= r )
proofend;

theorem Th48: :: BHSP_2:48
for X being RealUnitarySpace
for z, x being Point of X
for r being Real holds
( z in cl_Ball (x,r) iff dist (x,z) <= r )
proofend;

theorem :: BHSP_2:49
for X being RealUnitarySpace
for x being Point of X
for r being Real st r >= 0 holds
x in cl_Ball (x,r)
proofend;

theorem Th50: :: BHSP_2:50
for X being RealUnitarySpace
for y, x being Point of X
for r being Real st y in Ball (x,r) holds
y in cl_Ball (x,r)
proofend;

theorem Th51: :: BHSP_2:51
for X being RealUnitarySpace
for z, x being Point of X
for r being Real holds
( z in Sphere (x,r) iff ||.(x - z).|| = r )
proofend;

theorem :: BHSP_2:52
for X being RealUnitarySpace
for z, x being Point of X
for r being Real holds
( z in Sphere (x,r) iff dist (x,z) = r )
proofend;

theorem :: BHSP_2:53
for X being RealUnitarySpace
for y, x being Point of X
for r being Real st y in Sphere (x,r) holds
y in cl_Ball (x,r)
proofend;

theorem Th54: :: BHSP_2:54
for X being RealUnitarySpace
for x being Point of X
for r being Real holds Ball (x,r) c= cl_Ball (x,r)
proofend;

theorem Th55: :: BHSP_2:55
for X being RealUnitarySpace
for x being Point of X
for r being Real holds Sphere (x,r) c= cl_Ball (x,r)
proofend;

theorem :: BHSP_2:56
for X being RealUnitarySpace
for x being Point of X
for r being Real holds (Ball (x,r)) \/ (Sphere (x,r)) = cl_Ball (x,r)
proofend;