:: Introduction to Banach and Hilbert spaces - Part II
:: by Jan Popio{\l}ek
::
:: Received July 19, 1991
:: Copyright (c) 1991 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;
attr seq 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 V11() holds
seq is convergent

proof end;

theorem Th2: :: BHSP_2:2

for X being RealUnitarySpace
for seq, seq9 being sequence of X st seq is convergent & 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

proof end;

theorem Th3: :: BHSP_2:3

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

proof end;

theorem Th4: :: BHSP_2:4

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

proof end;

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

proof end;

theorem Th6: :: BHSP_2:6

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

proof end;

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

proof end;

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

proof end;

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 )

proof end;

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 ) & ( 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

proof end;

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 V11() & x in rng seq holds
lim seq = x

proof end;

theorem :: BHSP_2:11

for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X st seq is V11() & ex n being Element of NAT st seq . n = x holds
lim seq = x

proof end;

theorem :: BHSP_2:12

for X being RealUnitarySpace
for seq, seq9 being sequence of X st seq is convergent & 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

proof end;

theorem Th13: :: BHSP_2:13

for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds
lim (seq1 + seq2) = (lim seq1) + (lim seq2)

proof end;

theorem Th14: :: BHSP_2:14

for X being RealUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds
lim (seq1 - seq2) = (lim seq1) - (lim seq2)

proof end;

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)

proof end;

theorem Th16: :: BHSP_2:16

for X being RealUnitarySpace
for seq being sequence of X st seq is convergent holds
lim (- seq) = - (lim seq)

proof end;

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

proof end;

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

proof end;

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 )

proof end;

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).||

proof end;

uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Element of NAT holds b₁ . n = ||.(seq . n).|| ) & ( for n being Element of NAT holds b₂ . n = ||.(seq . n).|| ) holds
b₁ = b₂

proof end;

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

proof end;

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 & lim seq = g holds
( ||.seq.|| is convergent & lim ||.seq.|| = ||.g.|| )

proof end;

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 & lim seq = g holds
( ||.(seq - g).|| is convergent & lim ||.(seq - g).|| = 0 )

proof end;

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)

proof end;

uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Element of NAT holds b₁ . n = dist ((seq . n),x) ) & ( for n being Element of NAT holds b₂ . n = dist ((seq . n),x) ) holds
b₁ = b₂

proof end;

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 & lim seq = g holds
dist (seq,g) is convergent

proof end;

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 & lim seq = g holds
( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 )

proof end;

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 & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.(seq1 + seq2).|| is convergent & lim ||.(seq1 + seq2).|| = ||.(g1 + g2).|| )

proof end;

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 & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )

proof end;

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 & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.(seq1 - seq2).|| is convergent & lim ||.(seq1 - seq2).|| = ||.(g1 - g2).|| )

proof end;

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 & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.((seq1 - seq2) - (g1 - g2)).|| is convergent & lim ||.((seq1 - seq2) - (g1 - g2)).|| = 0 )

proof end;

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 & lim seq = g holds
( ||.(a * seq).|| is convergent & lim ||.(a * seq).|| = ||.(a * g).|| )

proof end;

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 & lim seq = g holds
( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

proof end;

theorem :: BHSP_2:31

for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.(- seq).|| is convergent & lim ||.(- seq).|| = ||.(- g).|| )

proof end;

theorem :: BHSP_2:32

for X being RealUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.((- seq) - (- g)).|| is convergent & lim ||.((- seq) - (- g)).|| = 0 )

proof end;

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

proof end;

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 & lim seq = g holds
( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )

proof end;

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 & lim seq = g holds
( ||.(seq - x).|| is convergent & lim ||.(seq - x).|| = ||.(g - x).|| )

proof end;

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 & lim seq = g holds
( ||.((seq - x) - (g - x)).|| is convergent & lim ||.((seq - x) - (g - x)).|| = 0 )

proof end;

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 & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( dist ((seq1 + seq2),(g1 + g2)) is convergent & lim (dist ((seq1 + seq2),(g1 + g2))) = 0 )

proof end;

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 & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( dist ((seq1 - seq2),(g1 - g2)) is convergent & lim (dist ((seq1 - seq2),(g1 - g2))) = 0 )

proof end;

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 & lim seq = g holds
( dist ((a * seq),(a * g)) is convergent & lim (dist ((a * seq),(a * g))) = 0 )

proof end;

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 & lim seq = g holds
( dist ((seq + x),(g + x)) is convergent & lim (dist ((seq + x),(g + x))) = 0 )

proof end;

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

proof end;

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

proof end;

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

proof end;

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 )

proof end;

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 )

proof end;

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)

proof end;

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) & z in Ball (x,r) holds
dist (y,z) < 2 * r

proof end;

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)

proof end;

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)

proof end;

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) & r <= q holds
y in Ball (x,q)

proof end;

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 )

proof end;

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 )

proof end;

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)

proof end;

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)

proof end;

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 )

proof end;

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 )

proof end;

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)

proof end;

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)

proof end;

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)

proof end;

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)

proof end;