NORMSP_0: Preliminaries to Normed Spaces

:: Preliminaries to Normed Spaces
:: by Andrzej Trybulec
::
:: Received March 23, 2010
:: Copyright (c) 2010-2016 Association of Mizar Users

environ

vocabularies HIDDEN, SUBSET_1, STRUCT_0, NORMSP_1, FUNCT_1, NUMBERS, REAL_1, XBOOLE_0, FUNCT_5, CARD_1, METRIC_1, RELAT_2, SUPINF_2, XCMPLX_0, NAT_1, SEQ_1, RELAT_1, TARSKI, PARTFUN1, NORMSP_0;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, FUNCOP_1, FUNCT_5, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, NAT_1, SEQ_1, STRUCT_0;
definitions TARSKI, RELAT_1, FUNCT_1;
theorems FUNCOP_1, TARSKI, FUNCT_1, RELAT_1, RELSET_1, PARTFUN1, FUNCT_2, STRUCT_0, XREAL_0, ORDINAL1;
schemes FUNCT_1;
registrations STRUCT_0, RELSET_1, XBOOLE_0, NUMBERS, XREAL_0, ORDINAL1, FUNCT_2;
constructors HIDDEN, FUNCT_2, NUMBERS, STRUCT_0, FUNCT_5, FUNCOP_1, XCMPLX_0, RELSET_1, XREAL_0;
requirements HIDDEN, BOOLE, SUBSET;
equalities FUNCT_5, FUNCOP_1, STRUCT_0;
expansions ;

definition

let RNS be non empty 1-sorted ;
let s be sequence of RNS;
let n be Nat;
:: original: .
redefine func s . n -> Element of RNS;
coherence

proof end;

end;

definition

attr c₁ is strict ;
struct N-Str -> 1-sorted ;
aggr N-Str(# carrier, normF #) -> N-Str ;
sel normF c₁ -> Function of the carrier of c₁,REAL;

end;

0 in REAL
by XREAL_0:def 1;

then reconsider f = op1 as Function of {0},REAL by FUNCOP_1:46;

reconsider z = 0 as Element of {0} by TARSKI:def 1;

registration

cluster non empty strict for N-Str ;

proof end;

end;

definition

let X be non empty N-Str ;
let x be Element of X;

func ||.x.|| -> Real equals :: NORMSP_0:def 1
the normF of X . x;

end;

:: deftheorem defines ||. NORMSP_0:def 1 :

definition

let X be non empty N-Str ;
let f be the carrier of X -valued Function;

func ||.f.|| -> Function means :Def2: :: NORMSP_0:def 2
( dom it = dom f & ( for e being set st e in dom it holds
it . e = ||.(f /. e).|| ) );

proof end;

proof end;

end;

:: deftheorem Def2 defines ||. NORMSP_0:def 2 :

registration

let X be non empty N-Str ;
let f be the carrier of X -valued Function;

cluster ||.f.|| -> REAL -valued ;

proof end;

end;

definition

let C be non empty set ;
let X be non empty N-Str ;
let f be PartFunc of C, the carrier of X;

:: original: ||.
redefine func ||.f.|| -> PartFunc of C,REAL means :: NORMSP_0:def 3
( dom it = dom f & ( for c being Element of C st c in dom it holds
it . c = ||.(f /. c).|| ) );

proof end;

proof end;

end;

:: deftheorem defines ||. NORMSP_0:def 3 :

definition

let X be non empty N-Str ;
let s be sequence of X;

:: original: ||.
redefine func ||.s.|| -> Real_Sequence means :: NORMSP_0:def 4
for n being Nat holds it . n = ||.(s . n).||;

proof end;

proof end;

end;

:: deftheorem defines ||. NORMSP_0:def 4 :

definition

attr c₁ is strict ;
struct N-ZeroStr -> N-Str , ZeroStr ;
aggr N-ZeroStr(# carrier, ZeroF, normF #) -> N-ZeroStr ;

end;

registration

cluster non empty strict for N-ZeroStr ;

proof end;

end;

definition

let X be non empty N-ZeroStr ;

attr X is discerning means :: NORMSP_0:def 5
for x being Element of X st ||.x.|| = 0 holds
x = 0. X;

attr X is reflexive means :Def6: :: NORMSP_0:def 6
||.(0. X).|| = 0 ;

end;

:: deftheorem defines discerning NORMSP_0:def 5 :

:: deftheorem Def6 defines reflexive NORMSP_0:def 6 :

registration

cluster non empty strict discerning reflexive for N-ZeroStr ;

proof end;

end;

registration

let X be non empty reflexive N-ZeroStr ;
let x be zero Element of X;

cluster ||.x.|| -> zero ;

proof end;

end;