BHSP_1: Introduction to Banach and Hilbert spaces

:: Introduction to Banach and Hilbert spaces - Part I
:: by Jan Popio{\l}ek
::
:: Received July 19, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

definition

attr c₁ is strict ;
struct UNITSTR -> RLSStruct ;
aggr UNITSTR(# carrier, ZeroF, addF, Mult, scalar #) -> UNITSTR ;
sel scalar c₁ -> Function of [: the carrier of c₁, the carrier of c₁:],REAL;

end;

registration

cluster non empty strict for UNITSTR ;

existence

proof end;

end;

registration

let D be non empty set ;
let Z be Element of D;
let a be BinOp of D;
let m be Function of [:REAL,D:],D;
let s be Function of [:D,D:],REAL;

cluster UNITSTR(# D,Z,a,m,s #) -> non empty ;

coherence ;

end;

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

definition

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

func x .|. y -> Real equals :: BHSP_1:def 1
the scalar of X . [x,y];

correctness ;

end;

:: deftheorem defines .|. BHSP_1:def 1 :

set V0 = the RealLinearSpace;

Lm1: the carrier of ((0). the RealLinearSpace) = {(0. the RealLinearSpace)}
by RLSUB_1:def 3;

reconsider nilfunc = [: the carrier of ((0). the RealLinearSpace), the carrier of ((0). the RealLinearSpace):] --> (In (0,REAL)) as Function of [: the carrier of ((0). the RealLinearSpace), the carrier of ((0). the RealLinearSpace):],REAL ;

( ( for x, y being VECTOR of ((0). the RealLinearSpace) holds nilfunc . [x,y] = 0 ) & 0. the RealLinearSpace in the carrier of ((0). the RealLinearSpace) )
by Lm1, FUNCOP_1:7, TARSKI:def 1;

then Lm2: nilfunc . [(0. the RealLinearSpace),(0. the RealLinearSpace)] = 0
;

Lm3: for u, v being VECTOR of ((0). the RealLinearSpace) holds nilfunc . [u,v] = nilfunc . [v,u]

proof end;

Lm4: for u, v, w being VECTOR of ((0). the RealLinearSpace) holds nilfunc . [(u + v),w] = (nilfunc . [u,w]) + (nilfunc . [v,w])

proof end;

Lm5: for u, v being VECTOR of ((0). the RealLinearSpace)
for a being Real holds nilfunc . [(a * u),v] = a * (nilfunc . [u,v])

proof end;

set X0 = UNITSTR(# the carrier of ((0). the RealLinearSpace),(0. ((0). the RealLinearSpace)), the addF of ((0). the RealLinearSpace), the Mult of ((0). the RealLinearSpace),nilfunc #);

Lm6: now :: thesis:

proof

reconsider u = x, v = y, w = z as VECTOR of ((0). the RealLinearSpace) ;
(x + y) .|. z = nilfunc . [(u + v),w] ;
hence (x + y) .|. z = (x .|. z) + (y .|. z) by Lm4; :: thesis:

end;

thus (a * x) .|. y = a * (x .|. y) :: thesis:

proof

reconsider u = x, v = y as VECTOR of ((0). the RealLinearSpace) ;
(a * x) .|. y = nilfunc . [(a * u),v] ;
hence (a * x) .|. y = a * (x .|. y) by Lm5; :: thesis:

end;

definition

let IT be non empty UNITSTR ;

attr IT is RealUnitarySpace-like means :Def2: :: BHSP_1:def 2
for x, y, z being Point of IT
for a being Real holds
( ( x .|. x = 0 implies x = 0. IT ) & ( x = 0. IT implies x .|. x = 0 ) & 0 <= x .|. x & x .|. y = y .|. x & (x + y) .|. z = (x .|. z) + (y .|. z) & (a * x) .|. y = a * (x .|. y) );

end;

:: deftheorem Def2 defines RealUnitarySpace-like BHSP_1:def 2 :

registration

cluster non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like for UNITSTR ;

existence

proof end;

end;

definition

mode RealUnitarySpace is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like UNITSTR ;

end;

definition

let X be RealUnitarySpace;
let x, y be Point of X;
:: original: .|.
redefine func x .|. y -> Real;
commutativity by Def2;

end;

theorem :: BHSP_1:1

for X being RealUnitarySpace holds (0. X) .|. (0. X) = 0 by Def2;

theorem :: BHSP_1:2

for X being RealUnitarySpace
for x, y, z being Point of X holds x .|. (y + z) = (x .|. y) + (x .|. z) by Def2;

theorem :: BHSP_1:3

for a being Real
for X being RealUnitarySpace
for x, y being Point of X holds x .|. (a * y) = a * (x .|. y) by Def2;

theorem :: BHSP_1:4

for a being Real
for X being RealUnitarySpace
for x, y being Point of X holds (a * x) .|. y = x .|. (a * y)

proof end;

theorem Th5: :: BHSP_1:5

for a, b being Real
for X being RealUnitarySpace
for x, y, z being Point of X holds ((a * x) + (b * y)) .|. z = (a * (x .|. z)) + (b * (y .|. z))

proof end;

theorem :: BHSP_1:6

for a, b being Real
for X being RealUnitarySpace
for x, y, z being Point of X holds x .|. ((a * y) + (b * z)) = (a * (x .|. y)) + (b * (x .|. z)) by Th5;

theorem :: BHSP_1:7

for X being RealUnitarySpace
for x, y being Point of X holds (- x) .|. y = x .|. (- y)

proof end;

theorem Th8: :: BHSP_1:8

for X being RealUnitarySpace
for x, y being Point of X holds (- x) .|. y = - (x .|. y)

proof end;

theorem :: BHSP_1:9

for X being RealUnitarySpace
for x, y being Point of X holds x .|. (- y) = - (x .|. y) by Th8;

theorem Th10: :: BHSP_1:10

for X being RealUnitarySpace
for x, y being Point of X holds (- x) .|. (- y) = x .|. y

proof end;

theorem Th11: :: BHSP_1:11

for X being RealUnitarySpace
for x, y, z being Point of X holds (x - y) .|. z = (x .|. z) - (y .|. z)

proof end;

theorem Th12: :: BHSP_1:12

for X being RealUnitarySpace
for x, y, z being Point of X holds x .|. (y - z) = (x .|. y) - (x .|. z)

proof end;

theorem :: BHSP_1:13

for X being RealUnitarySpace
for x, y, u, v being Point of X holds (x - y) .|. (u - v) = (((x .|. u) - (x .|. v)) - (y .|. u)) + (y .|. v)

proof end;

theorem Th14: :: BHSP_1:14

for X being RealUnitarySpace
for x being Point of X holds (0. X) .|. x = 0

proof end;

theorem :: BHSP_1:15

for X being RealUnitarySpace
for x being Point of X holds x .|. (0. X) = 0 by Th14;

theorem Th16: :: BHSP_1:16

for X being RealUnitarySpace
for x, y being Point of X holds (x + y) .|. (x + y) = ((x .|. x) + (2 * (x .|. y))) + (y .|. y)

proof end;

theorem :: BHSP_1:17

for X being RealUnitarySpace
for x, y being Point of X holds (x + y) .|. (x - y) = (x .|. x) - (y .|. y)

proof end;

theorem Th18: :: BHSP_1:18

for X being RealUnitarySpace
for x, y being Point of X holds (x - y) .|. (x - y) = ((x .|. x) - (2 * (x .|. y))) + (y .|. y)

proof end;

theorem Th19: :: BHSP_1:19

for X being RealUnitarySpace
for x, y being Point of X holds |.(x .|. y).| <= (sqrt (x .|. x)) * (sqrt (y .|. y))

proof end;

definition

let X be RealUnitarySpace;
let x, y be Point of X;

pred x,y are_orthogonal means :: BHSP_1:def 3
x .|. y = 0 ;

symmetry ;

end;

:: deftheorem defines are_orthogonal BHSP_1:def 3 :

theorem :: BHSP_1:20

for X being RealUnitarySpace
for x, y being Point of X st x,y are_orthogonal holds
x, - y are_orthogonal

proof end;

theorem :: BHSP_1:21

for X being RealUnitarySpace
for x, y being Point of X st x,y are_orthogonal holds
- x,y are_orthogonal

proof end;

theorem :: BHSP_1:22

for X being RealUnitarySpace
for x, y being Point of X st x,y are_orthogonal holds
- x, - y are_orthogonal by Th10;

theorem :: BHSP_1:23

for X being RealUnitarySpace
for x being Point of X holds x, 0. X are_orthogonal by Th14;

theorem :: BHSP_1:24

for X being RealUnitarySpace
for x, y being Point of X st x,y are_orthogonal holds
(x + y) .|. (x + y) = (x .|. x) + (y .|. y)

proof end;

theorem :: BHSP_1:25

for X being RealUnitarySpace
for x, y being Point of X st x,y are_orthogonal holds
(x - y) .|. (x - y) = (x .|. x) + (y .|. y)

proof end;

definition

let X be RealUnitarySpace;
let x be Point of X;

func ||.x.|| -> Real equals :: BHSP_1:def 4
sqrt (x .|. x);

correctness ;

end;

:: deftheorem defines ||. BHSP_1:def 4 :

theorem Th26: :: BHSP_1:26

for X being RealUnitarySpace
for x being Point of X holds
( ||.x.|| = 0 iff x = 0. X )

proof end;

theorem Th27: :: BHSP_1:27

for a being Real
for X being RealUnitarySpace
for x being Point of X holds ||.(a * x).|| = |.a.| * ||.x.||

proof end;

theorem Th28: :: BHSP_1:28

for X being RealUnitarySpace
for x being Point of X holds 0 <= ||.x.||

proof end;

theorem :: BHSP_1:29

for X being RealUnitarySpace
for x, y being Point of X holds |.(x .|. y).| <= ||.x.|| * ||.y.|| by Th19;

theorem Th30: :: BHSP_1:30

for X being RealUnitarySpace
for x, y being Point of X holds ||.(x + y).|| <= ||.x.|| + ||.y.||

proof end;

theorem Th31: :: BHSP_1:31

for X being RealUnitarySpace
for x being Point of X holds ||.(- x).|| = ||.x.||

proof end;

theorem Th32: :: BHSP_1:32

for X being RealUnitarySpace
for x, y being Point of X holds ||.x.|| - ||.y.|| <= ||.(x - y).||

proof end;

theorem :: BHSP_1:33

for X being RealUnitarySpace
for x, y being Point of X holds |.(||.x.|| - ||.y.||).| <= ||.(x - y).||

proof end;

definition

let X be RealUnitarySpace;
let x, y be Point of X;

func dist (x,y) -> Real equals :: BHSP_1:def 5
||.(x - y).||;

correctness ;
commutativity

proof end;

end;

:: deftheorem defines dist BHSP_1:def 5 :

theorem Th34: :: BHSP_1:34

for X being RealUnitarySpace
for x being Point of X holds dist (x,x) = 0

proof end;

theorem :: BHSP_1:35

for X being RealUnitarySpace
for x, y, z being Point of X holds dist (x,z) <= (dist (x,y)) + (dist (y,z))

proof end;

theorem Th36: :: BHSP_1:36

for X being RealUnitarySpace
for x, y being Point of X holds
( x <> y iff dist (x,y) <> 0 )

proof end;

theorem :: BHSP_1:37

for X being RealUnitarySpace
for x, y being Point of X holds dist (x,y) >= 0 by Th28;

theorem :: BHSP_1:38

for X being RealUnitarySpace
for x, y being Point of X holds
( x <> y iff dist (x,y) > 0 )

proof end;

theorem :: BHSP_1:39

for X being RealUnitarySpace
for x, y being Point of X holds dist (x,y) = sqrt ((x - y) .|. (x - y)) ;

theorem :: BHSP_1:40

for X being RealUnitarySpace
for x, y, u, v being Point of X holds dist ((x + y),(u + v)) <= (dist (x,u)) + (dist (y,v))

proof end;

theorem :: BHSP_1:41

for X being RealUnitarySpace
for x, y, u, v being Point of X holds dist ((x - y),(u - v)) <= (dist (x,u)) + (dist (y,v))

proof end;

theorem :: BHSP_1:42

for X being RealUnitarySpace
for x, y, z being Point of X holds dist ((x - z),(y - z)) = dist (x,y)

proof end;

theorem :: BHSP_1:43

for X being RealUnitarySpace
for x, y, z being Point of X holds dist ((x - z),(y - z)) <= (dist (z,x)) + (dist (z,y))

proof end;

definition

let X be non empty addLoopStr ;
let seq be sequence of X;
let x be Point of X;

func seq + x -> sequence of X means :Def6: :: BHSP_1:def 6
for n being Nat holds it . n = (seq . n) + x;

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines + BHSP_1:def 6 :

theorem Th44: :: BHSP_1:44

for n being Nat
for X being non empty addLoopStr
for seq being sequence of X holds (- seq) . n = - (seq . n)

proof end;

definition

let X be RealUnitarySpace;
let seq1, seq2 be sequence of X;
:: original: +
redefine func seq1 + seq2 -> Element of bool [:NAT, the carrier of X:];
commutativity

proof end;

end;

theorem :: BHSP_1:45

for X being RealUnitarySpace
for seq1, seq2, seq3 being sequence of X holds seq1 + (seq2 + seq3) = (seq1 + seq2) + seq3

proof end;

theorem :: BHSP_1:46

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

proof end;

theorem :: BHSP_1:47

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

proof end;

theorem :: BHSP_1:48

for a being Real
for X being RealUnitarySpace
for seq1 being sequence of X st seq1 is constant holds
a * seq1 is constant

proof end;

theorem :: BHSP_1:49

for X being RealUnitarySpace
for seq1, seq2 being sequence of X holds seq1 - seq2 = seq1 + (- seq2)

proof end;

theorem :: BHSP_1:50

for X being RealUnitarySpace
for seq being sequence of X holds seq = seq + (0. X)

proof end;

theorem :: BHSP_1:51

for a being Real
for X being RealUnitarySpace
for seq1, seq2 being sequence of X holds a * (seq1 + seq2) = (a * seq1) + (a * seq2)

proof end;

theorem :: BHSP_1:52

for a, b being Real
for X being RealUnitarySpace
for seq being sequence of X holds (a + b) * seq = (a * seq) + (b * seq)

proof end;

theorem :: BHSP_1:53

for a, b being Real
for X being RealUnitarySpace
for seq being sequence of X holds (a * b) * seq = a * (b * seq)

proof end;

theorem :: BHSP_1:54

for X being RealUnitarySpace
for seq being sequence of X holds 1 * seq = seq

proof end;

theorem :: BHSP_1:55

for X being RealUnitarySpace
for seq being sequence of X holds (- 1) * seq = - seq

proof end;

theorem :: BHSP_1:56

for X being RealUnitarySpace
for x being Point of X
for seq being sequence of X holds seq - x = seq + (- x)

proof end;

theorem :: BHSP_1:57

for X being RealUnitarySpace
for seq1, seq2 being sequence of X holds seq1 - seq2 = - (seq2 - seq1)

proof end;

theorem :: BHSP_1:58

for X being RealUnitarySpace
for seq being sequence of X holds seq = seq - (0. X)

proof end;

theorem :: BHSP_1:59

for X being RealUnitarySpace
for seq being sequence of X holds seq = - (- seq)

proof end;

theorem :: BHSP_1:60

for X being RealUnitarySpace
for seq1, seq2, seq3 being sequence of X holds seq1 - (seq2 + seq3) = (seq1 - seq2) - seq3

proof end;

theorem :: BHSP_1:61

for X being RealUnitarySpace
for seq1, seq2, seq3 being sequence of X holds (seq1 + seq2) - seq3 = seq1 + (seq2 - seq3)

proof end;

theorem :: BHSP_1:62

for X being RealUnitarySpace
for seq1, seq2, seq3 being sequence of X holds seq1 - (seq2 - seq3) = (seq1 - seq2) + seq3

proof end;

theorem :: BHSP_1:63

for a being Real
for X being RealUnitarySpace
for seq1, seq2 being sequence of X holds a * (seq1 - seq2) = (a * seq1) - (a * seq2)

proof end;