:: Real Normed Space
:: by Jan Popio{\l}ek
::
:: Received September 20, 1990
:: Copyright (c) 1990-2011 Association of Mizar Users


begin

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

registration
cluster non empty strict 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) --> 0 as Function of the carrier of ((0). the RealLinearSpace),REAL by FUNCOP_1:57;

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

then Lm2: niltonil . (0. the RealLinearSpace) = 0
by FUNCOP_1:13;

Lm3: for u being VECTOR of ((0). the RealLinearSpace)
for a being Real holds niltonil . (a * u) = (abs 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
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).|| = (abs 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).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
reconsider u = x, w = y as VECTOR of ((0). the RealLinearSpace) ;
H1(X) = 0. the RealLinearSpace by RLSUB_1:19;
hence ( ||.x.|| = 0 iff x = H1(X) ) by Lm1, FUNCOP_1:13, TARSKI:def 1; :: thesis: ( ||.(a * x).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
a * x = a * u ;
hence ||.(a * x).|| = (abs a) * ||.x.|| by Lm3; :: thesis: ||.(x + y).|| <= ||.x.|| + ||.y.||
x + y = u + w ;
hence ||.(x + y).|| <= ||.x.|| + ||.y.|| by Lm4; :: thesis: verum
end;

definition
canceled;
let IT be non empty NORMSTR ;
attr IT is RealNormSpace-like means :Def2: :: NORMSP_1:def 2
 for x, y being Point of IT
for a being Real holds
( ||.(a * x).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| );
end;

:: deftheorem NORMSP_1:def 1 :
canceled;

:: deftheorem Def2 defines RealNormSpace-like NORMSP_1:def 2 :
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).|| = (abs 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 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
canceled;

theorem :: NORMSP_1:2
canceled;

theorem :: NORMSP_1:3
canceled;

theorem :: NORMSP_1:4
canceled;

theorem :: NORMSP_1:5
for RNS being RealNormSpace holds ||.(0. RNS).|| = 0 ;

theorem Th6: :: NORMSP_1:6
for RNS being RealNormSpace
for x being Point of RNS holds ||.(- x).|| = ||.x.||
proof end;

theorem Th7: :: NORMSP_1:7
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.(x - y).|| <= ||.x.|| + ||.y.||
proof end;

theorem Th8: :: NORMSP_1:8
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 Th8;
end;

theorem :: NORMSP_1:9
for a, b being Real
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.((a * x) + (b * y)).|| <= ((abs a) * ||.x.||) + ((abs b) * ||.y.||)
proof end;

theorem Th10: :: NORMSP_1:10
for RNS being RealNormSpace
for x, y being Point of RNS holds
( ||.(x - y).|| = 0 iff x = y )
proof end;

theorem Th11: :: NORMSP_1:11
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.(x - y).|| = ||.(y - x).||
proof end;

theorem Th12: :: NORMSP_1:12
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.x.|| - ||.y.|| <= ||.(x - y).||
proof end;

theorem Th13: :: NORMSP_1:13
for RNS being RealNormSpace
for x, y being Point of RNS holds abs (||.x.|| - ||.y.||) <= ||.(x - y).||
proof end;

theorem Th14: :: NORMSP_1:14
for RNS being RealNormSpace
for x, z, y being Point of RNS holds ||.(x - z).|| <= ||.(x - y).|| + ||.(y - z).||
proof end;

theorem :: NORMSP_1:15
for RNS being RealNormSpace
for x, y being Point of RNS st x <> y holds
||.(x - y).|| <> 0 by Th10;

theorem :: NORMSP_1:16
canceled;

theorem :: NORMSP_1:17
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_1:18
canceled;

theorem :: NORMSP_1:19
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_1:20
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_1:21
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 Element of NAT holds S . n = S . (n + 1)
proof end;

theorem :: NORMSP_1:22
for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n being Element of NAT holds S . n = S . (n + 1) ) holds
for n, k being Element of NAT holds S . n = S . (n + k)
proof end;

theorem :: NORMSP_1:23
for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n, k being Element of NAT holds S . n = S . (n + k) ) holds
for n, m being Element of NAT holds S . n = S . m
proof end;

theorem :: NORMSP_1:24
for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n, m being Element of NAT holds S . n = S . m ) holds
ex x being Element of RNS st
for n being Nat holds S . n = x
proof end;

theorem :: NORMSP_1:25
canceled;

theorem :: NORMSP_1:26
canceled;

theorem :: NORMSP_1:27
canceled;

Lm6: for RNS being non empty 1-sorted
for S being sequence of RNS
for n being Element of NAT holds S . n is Element of RNS
;

definition
canceled;
let RNS be non empty 1-sorted ;
let S be sequence of RNS;
let n be Element of NAT ;
:: original: .
redefine func S . n -> Element of RNS;
coherence
S . n is Element of RNS by Lm6;
end;

:: deftheorem NORMSP_1:def 3 :
canceled;

definition
let RNS be non empty addLoopStr ;
let S1, S2 be sequence of RNS;
canceled;
func S1 + S2 -> sequence of RNS means :Def5: :: NORMSP_1:def 5
 for n being Element of NAT holds it . n = (S1 . n) + (S2 . n);
existence
ex b1 being sequence of RNS st
for n being Element of NAT holds b1 . n = (S1 . n) + (S2 . n)
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Element of NAT holds b1 . n = (S1 . n) + (S2 . n) ) & ( for n being Element of NAT holds b2 . n = (S1 . n) + (S2 . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem NORMSP_1:def 4 :
canceled;

:: deftheorem Def5 defines + NORMSP_1:def 5 :
for RNS being non empty addLoopStr
for S1, S2, b4 being sequence of RNS holds
( b4 = S1 + S2 iff for n being Element of 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 :Def6: :: NORMSP_1:def 6
 for n being Element of NAT holds it . n = (S1 . n) - (S2 . n);
existence
ex b1 being sequence of RNS st
for n being Element of NAT holds b1 . n = (S1 . n) - (S2 . n)
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Element of NAT holds b1 . n = (S1 . n) - (S2 . n) ) & ( for n being Element of NAT holds b2 . n = (S1 . n) - (S2 . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines - NORMSP_1:def 6 :
for RNS being non empty addLoopStr
for S1, S2, b4 being sequence of RNS holds
( b4 = S1 - S2 iff for n being Element of 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 :Def7: :: NORMSP_1:def 7
 for n being Element of NAT holds it . n = (S . n) - x;
existence
ex b1 being sequence of RNS st
for n being Element of NAT holds b1 . n = (S . n) - x
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Element of NAT holds b1 . n = (S . n) - x ) & ( for n being Element of NAT holds b2 . n = (S . n) - x ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines - NORMSP_1:def 7 :
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 Element of 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 :Def8: :: NORMSP_1:def 8
 for n being Element of NAT holds it . n = a * (S . n);
existence
ex b1 being sequence of RNS st
for n being Element of NAT holds b1 . n = a * (S . n)
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Element of NAT holds b1 . n = a * (S . n) ) & ( for n being Element of NAT holds b2 . n = a * (S . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines * NORMSP_1:def 8 :
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 Element of NAT holds b4 . n = a * (S . n) );

definition
let RNS be RealNormSpace;
let S be sequence of RNS;
attr S is convergent means :Def9: :: NORMSP_1:def 9
 ex g being Point of RNS st
for r being Real st 0 < r holds
ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.((S . n) - g).|| < r;
end;

:: deftheorem Def9 defines convergent NORMSP_1:def 9 :
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 Element of NAT st
for n being Element of NAT st m <= n holds
||.((S . n) - g).|| < r );

theorem :: NORMSP_1:28
canceled;

theorem :: NORMSP_1:29
canceled;

theorem :: NORMSP_1:30
canceled;

theorem :: NORMSP_1:31
canceled;

theorem :: NORMSP_1:32
canceled;

theorem :: NORMSP_1:33
canceled;

theorem Th34: :: NORMSP_1:34
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 Th35: :: NORMSP_1:35
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 Th36: :: NORMSP_1:36
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 Th37: :: NORMSP_1:37
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 :: NORMSP_1:38
canceled;

theorem Th39: :: NORMSP_1:39
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
||.S.|| is convergent
proof end;

definition
canceled;
let RNS be RealNormSpace;
let S be sequence of RNS;
assume A1: S is convergent ;
func lim S -> Point of RNS means :Def11: :: NORMSP_1:def 11
 for r being Real st 0 < r holds
ex m being Element of NAT st
for n being Element of 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 Element of NAT st
for n being Element of NAT st m <= n holds
||.((S . n) - b1).|| < r by A1, Def9;
uniqueness
for b1, b2 being Point of RNS st ( for r being Real st 0 < r holds
ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.((S . n) - b1).|| < r ) & ( for r being Real st 0 < r holds
ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.((S . n) - b2).|| < r ) holds
b1 = b2
proof end;
end;

:: deftheorem NORMSP_1:def 10 :
canceled;

:: deftheorem Def11 defines lim NORMSP_1:def 11 :
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 Element of NAT st
for n being Element of NAT st m <= n holds
||.((S . n) - b3).|| < r );

theorem :: NORMSP_1:40
canceled;

theorem :: NORMSP_1:41
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_1:42
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_1:43
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_1:44
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_1:45
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;