:: Real Normed Space
:: by Jan Popio{\l}ek
::
:: Received September 20, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

attr c₁ is strict ;
struct NORMSTR -> RLSStruct , N-ZeroStr ;
aggr NORMSTR(# carrier, ZeroF, addF, Mult, normF #) -> NORMSTR ;

end;

registration

cluster non empty strict for NORMSTR ;

existence
ex b₁ being NORMSTR st
( not b₁ is empty & b₁ is strict )

proof end;

end;

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

set V = the RealLinearSpace;

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

reconsider niltonil = the carrier of ((0). the RealLinearSpace) --> (In (0,REAL)) as Function of the carrier of ((0). the RealLinearSpace),REAL ;

0. the RealLinearSpace is VECTOR of ((0). the RealLinearSpace)
by Lm1, TARSKI:def 1;

then Lm2: niltonil . (0. the RealLinearSpace) = 0
by FUNCOP_1:7;

Lm3: for u being VECTOR of ((0). the RealLinearSpace)
for a being Real holds niltonil . (a * u) = |.a.| * (niltonil . u)

proof end;

Lm4: for u, v being VECTOR of ((0). the RealLinearSpace) holds niltonil . (u + v) <= (niltonil . u) + (niltonil . v)

proof end;

reconsider X = NORMSTR(# the carrier of ((0). the RealLinearSpace),(0. ((0). the RealLinearSpace)), the addF of ((0). the RealLinearSpace), the Mult of ((0). the RealLinearSpace),niltonil #) as non empty NORMSTR ;

Lm5: now :: thesis: for x, y being Point of X
for a being Real holds
( ( ||.x.|| = 0 implies x = H₁(X) ) & ( x = H₁(X) implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )

let x, y be Point of X; :: thesis: for a being Real holds
( ( ||.x.|| = 0 implies x = H₁(X) ) & ( x = H₁(X) implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
let a be Real; :: thesis: ( ( ||.x.|| = 0 implies x = H₁(X) ) & ( x = H₁(X) implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
reconsider u = x, w = y as VECTOR of ((0). the RealLinearSpace) ;
H₁(X) = 0. the RealLinearSpace by RLSUB_1:11;
hence ( ||.x.|| = 0 iff x = H₁(X) ) by Lm1, FUNCOP_1:7, TARSKI:def 1; :: thesis: ( ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
a * x = a * u ;
hence ||.(a * x).|| = |.a.| * ||.x.|| by Lm3; :: thesis: ||.(x + y).|| <= ||.x.|| + ||.y.||
x + y = u + w ;
hence ||.(x + y).|| <= ||.x.|| + ||.y.|| by Lm4; :: thesis: verum

end;

definition

let IT be non empty NORMSTR ;

attr IT is RealNormSpace-like means :Def1: :: NORMSP_1:def 1
for x, y being Point of IT
for a being Real holds
( ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| );

end;

:: deftheorem Def1 defines RealNormSpace-like NORMSP_1:def 1 :
for IT being non empty NORMSTR holds
( IT is RealNormSpace-like iff for x, y being Point of IT
for a being Real holds
( ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| ) );

registration

cluster non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive strict RealNormSpace-like for NORMSTR ;

existence
ex b₁ being non empty NORMSTR st
( b₁ is reflexive & b₁ is discerning & b₁ is RealNormSpace-like & b₁ is vector-distributive & b₁ is scalar-distributive & b₁ is scalar-associative & b₁ is scalar-unital & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is strict )

proof end;

end;

definition

mode RealNormSpace is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like NORMSTR ;

end;

theorem :: NORMSP_1:1

for RNS being RealNormSpace holds ||.(0. RNS).|| = 0 ;

theorem Th2: :: NORMSP_1:2

for RNS being RealNormSpace
for x being Point of RNS holds ||.(- x).|| = ||.x.||

proof end;

theorem Th3: :: NORMSP_1:3

for RNS being RealNormSpace
for x, y being Point of RNS holds ||.(x - y).|| <= ||.x.|| + ||.y.||

proof end;

theorem Th4: :: NORMSP_1:4

for RNS being RealNormSpace
for x being Point of RNS holds 0 <= ||.x.||

proof end;

registration

let RNS be RealNormSpace;
let x be Point of RNS;

cluster ||.x.|| -> non negative ;

coherence
not ||.x.|| is negative by Th4;

end;

theorem :: NORMSP_1:5

for a, b being Real
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.((a * x) + (b * y)).|| <= (|.a.| * ||.x.||) + (|.b.| * ||.y.||)

proof end;

theorem Th6: :: NORMSP_1:6

for RNS being RealNormSpace
for x, y being Point of RNS holds
( ||.(x - y).|| = 0 iff x = y )

proof end;

theorem Th7: :: NORMSP_1:7

for RNS being RealNormSpace
for x, y being Point of RNS holds ||.(x - y).|| = ||.(y - x).||

proof end;

theorem Th8: :: NORMSP_1:8

for RNS being RealNormSpace
for x, y being Point of RNS holds ||.x.|| - ||.y.|| <= ||.(x - y).||

proof end;

theorem Th9: :: NORMSP_1:9

for RNS being RealNormSpace
for x, y being Point of RNS holds |.(||.x.|| - ||.y.||).| <= ||.(x - y).||

proof end;

theorem Th10: :: NORMSP_1:10

for RNS being RealNormSpace
for x, y, z being Point of RNS holds ||.(x - z).|| <= ||.(x - y).|| + ||.(y - z).||

proof end;

theorem :: NORMSP_1:11

for RNS being RealNormSpace
for x, y being Point of RNS st x <> y holds
||.(x - y).|| <> 0 by Th6;

theorem :: NORMSP_1:12

for f being Function
for RNS being non empty 1-sorted
for x being Element of RNS holds
( f is sequence of RNS iff ( dom f = NAT & ( for d being set st d in NAT holds
f . d is Element of RNS ) ) )

proof end;

theorem :: NORMSP_1:13

for RNS being non empty 1-sorted
for x being Element of RNS ex S being sequence of RNS st rng S = {x}

proof end;

theorem :: NORMSP_1:14

for RNS being non empty 1-sorted
for S being sequence of RNS st ex x being Element of RNS st
for n being Nat holds S . n = x holds
ex x being Element of RNS st rng S = {x}

proof end;

theorem :: NORMSP_1:15

for RNS being non empty 1-sorted
for S being sequence of RNS st ex x being Element of RNS st rng S = {x} holds
for n being Nat holds S . n = S . (n + 1)

proof end;

theorem :: NORMSP_1:16

for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n being Nat holds S . n = S . (n + 1) ) holds
for n, k being Nat holds S . n = S . (n + k)

proof end;

theorem :: NORMSP_1:17

for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n, k being Nat holds S . n = S . (n + k) ) holds
for n, m being Nat holds S . n = S . m

proof end;

theorem :: NORMSP_1:18

for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n, m being Nat holds S . n = S . m ) holds
ex x being Element of RNS st
for n being Nat holds S . n = x

proof end;

definition

let RNS be non empty addLoopStr ;
let S1, S2 be sequence of RNS;

func S1 + S2 -> sequence of RNS means :Def2: :: NORMSP_1:def 2
for n being Nat holds it . n = (S1 . n) + (S2 . n);

existence
ex b₁ being sequence of RNS st
for n being Nat holds b₁ . n = (S1 . n) + (S2 . n)

proof end;

uniqueness
for b₁, b₂ being sequence of RNS st ( for n being Nat holds b₁ . n = (S1 . n) + (S2 . n) ) & ( for n being Nat holds b₂ . n = (S1 . n) + (S2 . n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines + NORMSP_1:def 2 :
for RNS being non empty addLoopStr
for S1, S2, b₄ being sequence of RNS holds
( b₄ = S1 + S2 iff for n being Nat holds b₄ . n = (S1 . n) + (S2 . n) );

definition

let RNS be non empty addLoopStr ;
let S1, S2 be sequence of RNS;

func S1 - S2 -> sequence of RNS means :Def3: :: NORMSP_1:def 3
for n being Nat holds it . n = (S1 . n) - (S2 . n);

existence
ex b₁ being sequence of RNS st
for n being Nat holds b₁ . n = (S1 . n) - (S2 . n)

proof end;

uniqueness
for b₁, b₂ being sequence of RNS st ( for n being Nat holds b₁ . n = (S1 . n) - (S2 . n) ) & ( for n being Nat holds b₂ . n = (S1 . n) - (S2 . n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines - NORMSP_1:def 3 :
for RNS being non empty addLoopStr
for S1, S2, b₄ being sequence of RNS holds
( b₄ = S1 - S2 iff for n being Nat holds b₄ . n = (S1 . n) - (S2 . n) );

definition

let RNS be non empty addLoopStr ;
let S be sequence of RNS;
let x be Element of RNS;

func S - x -> sequence of RNS means :Def4: :: NORMSP_1:def 4
for n being Nat holds it . n = (S . n) - x;

existence
ex b₁ being sequence of RNS st
for n being Nat holds b₁ . n = (S . n) - x

proof end;

uniqueness
for b₁, b₂ being sequence of RNS st ( for n being Nat holds b₁ . n = (S . n) - x ) & ( for n being Nat holds b₂ . n = (S . n) - x ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines - NORMSP_1:def 4 :
for RNS being non empty addLoopStr
for S being sequence of RNS
for x being Element of RNS
for b₄ being sequence of RNS holds
( b₄ = S - x iff for n being Nat holds b₄ . n = (S . n) - x );

definition

let RNS be non empty RLSStruct ;
let S be sequence of RNS;
let a be Real;

func a * S -> sequence of RNS means :Def5: :: NORMSP_1:def 5
for n being Nat holds it . n = a * (S . n);

existence
ex b₁ being sequence of RNS st
for n being Nat holds b₁ . n = a * (S . n)

proof end;

uniqueness
for b₁, b₂ being sequence of RNS st ( for n being Nat holds b₁ . n = a * (S . n) ) & ( for n being Nat holds b₂ . n = a * (S . n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines * NORMSP_1:def 5 :
for RNS being non empty RLSStruct
for S being sequence of RNS
for a being Real
for b₄ being sequence of RNS holds
( b₄ = a * S iff for n being Nat holds b₄ . n = a * (S . n) );

definition

let RNS be RealNormSpace;
let S be sequence of RNS;

attr S is convergent means :: NORMSP_1:def 6
ex g being Point of RNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - g).|| < r;

end;

:: deftheorem defines convergent NORMSP_1:def 6 :
for RNS being RealNormSpace
for S being sequence of RNS holds
( S is convergent iff ex g being Point of RNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - g).|| < r );

theorem Th19: :: NORMSP_1:19

for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
S1 + S2 is convergent

proof end;

theorem Th20: :: NORMSP_1:20

for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
S1 - S2 is convergent

proof end;

theorem Th21: :: NORMSP_1:21

for RNS being RealNormSpace
for x being Point of RNS
for S being sequence of RNS st S is convergent holds
S - x is convergent

proof end;

theorem Th22: :: NORMSP_1:22

for a being Real
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
a * S is convergent

proof end;

theorem Th23: :: NORMSP_1:23

for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
||.S.|| is convergent

proof end;

definition

let RNS be RealNormSpace;
let S be sequence of RNS;
assume A1: S is convergent ;

func lim S -> Point of RNS means :Def7: :: NORMSP_1:def 7
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - it).|| < r;

existence
ex b₁ being Point of RNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₁).|| < r by A1;
uniqueness
for b₁, b₂ being Point of RNS st ( for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₁).|| < r ) & ( for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₂).|| < r ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines lim NORMSP_1:def 7 :
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
for b₃ being Point of RNS holds
( b₃ = lim S iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b₃).|| < r );

theorem :: NORMSP_1:24

for RNS being RealNormSpace
for g being Point of RNS
for S being sequence of RNS st S is convergent & lim S = g holds
( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 )

proof end;

theorem :: NORMSP_1:25

for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
lim (S1 + S2) = (lim S1) + (lim S2)

proof end;

theorem :: NORMSP_1:26

for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
lim (S1 - S2) = (lim S1) - (lim S2)

proof end;

theorem :: NORMSP_1:27

for RNS being RealNormSpace
for x being Point of RNS
for S being sequence of RNS st S is convergent holds
lim (S - x) = (lim S) - x

proof end;

theorem :: NORMSP_1:28

for a being Real
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
lim (a * S) = a * (lim S)

proof end;