:: RSSPACE4 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

Lm1: for rseq being Real_Sequence
for K being real number st ( for n being Nat holds rseq . n <= K ) holds
sup (rng rseq) <= K
proof end;

Lm2: for rseq being Real_Sequence st rseq is bounded holds
for n being Nat holds rseq . n <= sup (rng rseq)
proof end;

definition
func the_set_of_BoundedRealSequences -> Subset of Linear_Space_of_RealSequences means :Def1: :: RSSPACE4:def 1
for x being set holds
( x in it iff ( x in the_set_of_RealSequences & seq_id x is bounded ) );
existence
ex b1 being Subset of Linear_Space_of_RealSequences st
for x being set holds
( x in b1 iff ( x in the_set_of_RealSequences & seq_id x is bounded ) )
proof end;
uniqueness
for b1, b2 being Subset of Linear_Space_of_RealSequences st ( for x being set holds
( x in b1 iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) ) & ( for x being set holds
( x in b2 iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines the_set_of_BoundedRealSequences RSSPACE4:def 1 :
for b1 being Subset of Linear_Space_of_RealSequences holds
( b1 = the_set_of_BoundedRealSequences iff for x being set holds
( x in b1 iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) );

registration
cluster the_set_of_BoundedRealSequences -> non empty ;
coherence
not the_set_of_BoundedRealSequences is empty
proof end;
cluster the_set_of_BoundedRealSequences -> lineary-closed ;
coherence
the_set_of_BoundedRealSequences is lineary-closed
proof end;
end;

Lm3: RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is Subspace of Linear_Space_of_RealSequences
by RSSPACE:13;

registration
cluster RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) -> Abelian add-associative right_zeroed right_complementable RealLinearSpace-like ;
coherence
( RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is Abelian & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is add-associative & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is right_zeroed & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is right_complementable & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is RealLinearSpace-like )
by RSSPACE:13;
end;

Lm4: ( RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is Abelian & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is add-associative & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is right_zeroed & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is right_complementable & RLSStruct(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ) #) is RealLinearSpace-like )
;

Lm5: ex NORM being Function of the_set_of_BoundedRealSequences , REAL st
for x being set st x in the_set_of_BoundedRealSequences holds
NORM . x = sup (rng (abs (seq_id x)))
proof end;

definition
func linfty_norm -> Function of the_set_of_BoundedRealSequences , REAL means :Def2: :: RSSPACE4:def 2
for x being set st x in the_set_of_BoundedRealSequences holds
it . x = sup (rng (abs (seq_id x)));
existence
ex b1 being Function of the_set_of_BoundedRealSequences , REAL st
for x being set st x in the_set_of_BoundedRealSequences holds
b1 . x = sup (rng (abs (seq_id x)))
by Lm5;
uniqueness
for b1, b2 being Function of the_set_of_BoundedRealSequences , REAL st ( for x being set st x in the_set_of_BoundedRealSequences holds
b1 . x = sup (rng (abs (seq_id x))) ) & ( for x being set st x in the_set_of_BoundedRealSequences holds
b2 . x = sup (rng (abs (seq_id x))) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines linfty_norm RSSPACE4:def 2 :
for b1 being Function of the_set_of_BoundedRealSequences , REAL holds
( b1 = linfty_norm iff for x being set st x in the_set_of_BoundedRealSequences holds
b1 . x = sup (rng (abs (seq_id x))) );

Lm6: for rseq being Real_Sequence st ( for n being Nat holds rseq . n = 0 ) holds
( rseq is bounded & sup (rng (abs rseq)) = 0 )
proof end;

Lm7: for rseq being Real_Sequence st rseq is bounded & sup (rng (abs rseq)) = 0 holds
for n being Nat holds rseq . n = 0
proof end;

theorem :: RSSPACE4:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RSSPACE4:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for rseq being Real_Sequence holds
( ( rseq is bounded & sup (rng (abs rseq)) = 0 ) iff for n being Nat holds rseq . n = 0 ) by Lm6, Lm7;

registration
cluster NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #) -> Abelian add-associative right_zeroed right_complementable RealLinearSpace-like ;
coherence
( NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #) is Abelian & NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #) is add-associative & NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #) is right_zeroed & NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #) is right_complementable & NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #) is RealLinearSpace-like )
by Lm4, RSSPACE3:4;
end;

definition
func linfty_Space -> non empty NORMSTR equals :: RSSPACE4:def 3
NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #);
coherence
NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #) is non empty NORMSTR
;
end;

:: deftheorem defines linfty_Space RSSPACE4:def 3 :
linfty_Space = NORMSTR(# the_set_of_BoundedRealSequences ,(Zero_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Add_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),(Mult_ the_set_of_BoundedRealSequences ,Linear_Space_of_RealSequences ),linfty_norm #);

theorem Th3: :: RSSPACE4:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
( the carrier of linfty_Space = the_set_of_BoundedRealSequences & ( for x being set holds
( x is VECTOR of linfty_Space iff ( x is Real_Sequence & seq_id x is bounded ) ) ) & 0. linfty_Space = Zeroseq & ( for u being VECTOR of linfty_Space holds u = seq_id u ) & ( for u, v being VECTOR of linfty_Space holds u + v = (seq_id u) + (seq_id v) ) & ( for r being Real
for u being VECTOR of linfty_Space holds r * u = r (#) (seq_id u) ) & ( for u being VECTOR of linfty_Space holds
( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of linfty_Space holds u - v = (seq_id u) - (seq_id v) ) & ( for v being VECTOR of linfty_Space holds seq_id v is bounded ) & ( for v being VECTOR of linfty_Space holds ||.v.|| = sup (rng (abs (seq_id v))) ) )
proof end;

theorem Th4: :: RSSPACE4:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for x, y being Point of linfty_Space
for a being Real holds
( ( ||.x.|| = 0 implies x = 0. linfty_Space ) & ( x = 0. linfty_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = (abs a) * ||.x.|| )
proof end;

registration
cluster linfty_Space -> non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like RealNormSpace-like ;
coherence
( linfty_Space is RealNormSpace-like & linfty_Space is RealLinearSpace-like & linfty_Space is Abelian & linfty_Space is add-associative & linfty_Space is right_zeroed & linfty_Space is right_complementable )
by Th4, NORMSP_1:def 2;
end;

theorem :: RSSPACE4:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for vseq being sequence of linfty_Space st vseq is CCauchy holds
vseq is convergent
proof end;

definition
let X be non empty set ;
let Y be RealNormSpace;
let IT be Function of X,the carrier of Y;
attr IT is bounded means :Def4: :: RSSPACE4:def 4
ex K being Real st
( 0 <= K & ( for x being Element of X holds ||.(IT . x).|| <= K ) );
end;

:: deftheorem Def4 defines bounded RSSPACE4:def 4 :
for X being non empty set
for Y being RealNormSpace
for IT being Function of X,the carrier of Y holds
( IT is bounded iff ex K being Real st
( 0 <= K & ( for x being Element of X holds ||.(IT . x).|| <= K ) ) );

theorem Th6: :: RSSPACE4:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f being Function of X,the carrier of Y st ( for x being Element of X holds f . x = 0. Y ) holds
f is bounded
proof end;

registration
let X be non empty set ;
let Y be RealNormSpace;
cluster bounded M5(X,the carrier of Y);
existence
ex b1 being Function of X,the carrier of Y st b1 is bounded
proof end;
end;

definition
let X be non empty set ;
let Y be RealNormSpace;
func BoundedFunctions X,Y -> Subset of (RealVectSpace X,Y) means :Def5: :: RSSPACE4:def 5
for x being set holds
( x in it iff x is bounded Function of X,the carrier of Y );
existence
ex b1 being Subset of (RealVectSpace X,Y) st
for x being set holds
( x in b1 iff x is bounded Function of X,the carrier of Y )
proof end;
uniqueness
for b1, b2 being Subset of (RealVectSpace X,Y) st ( for x being set holds
( x in b1 iff x is bounded Function of X,the carrier of Y ) ) & ( for x being set holds
( x in b2 iff x is bounded Function of X,the carrier of Y ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines BoundedFunctions RSSPACE4:def 5 :
for X being non empty set
for Y being RealNormSpace
for b3 being Subset of (RealVectSpace X,Y) holds
( b3 = BoundedFunctions X,Y iff for x being set holds
( x in b3 iff x is bounded Function of X,the carrier of Y ) );

registration
let X be non empty set ;
let Y be RealNormSpace;
cluster BoundedFunctions X,Y -> non empty ;
coherence
not BoundedFunctions X,Y is empty
proof end;
end;

theorem Th7: :: RSSPACE4:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace holds BoundedFunctions X,Y is lineary-closed
proof end;

theorem Th8: :: RSSPACE4:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace holds RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) is Subspace of RealVectSpace X,Y
proof end;

registration
let X be non empty set ;
let Y be RealNormSpace;
cluster RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) -> Abelian add-associative right_zeroed right_complementable RealLinearSpace-like ;
coherence
( RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) is Abelian & RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) is add-associative & RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) is right_zeroed & RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) is right_complementable & RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) is RealLinearSpace-like )
by Th8;
end;

definition
let X be non empty set ;
let Y be RealNormSpace;
func R_VectorSpace_of_BoundedFunctions X,Y -> RealLinearSpace equals :: RSSPACE4:def 6
RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #);
coherence
RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #) is RealLinearSpace
;
end;

:: deftheorem defines R_VectorSpace_of_BoundedFunctions RSSPACE4:def 6 :
for X being non empty set
for Y being RealNormSpace holds R_VectorSpace_of_BoundedFunctions X,Y = RLSStruct(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)) #);

registration
let X be non empty set ;
let Y be RealNormSpace;
cluster R_VectorSpace_of_BoundedFunctions X,Y -> strict ;
coherence
R_VectorSpace_of_BoundedFunctions X,Y is strict
;
end;

theorem :: RSSPACE4:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th10: :: RSSPACE4:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f', g', h' being bounded Function of X,the carrier of Y st f' = f & g' = g & h' = h holds
( h = f + g iff for x being Element of X holds h' . x = (f' . x) + (g' . x) )
proof end;

theorem Th11: :: RSSPACE4:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f', h' being bounded Function of X,the carrier of Y st f' = f & h' = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) )
proof end;

theorem Th12: :: RSSPACE4:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace holds 0. (R_VectorSpace_of_BoundedFunctions X,Y) = X --> (0. Y)
proof end;

definition
let X be non empty set ;
let Y be RealNormSpace;
let f be set ;
assume A1: f in BoundedFunctions X,Y ;
func modetrans f,X,Y -> bounded Function of X,the carrier of Y equals :Def7: :: RSSPACE4:def 7
f;
coherence
f is bounded Function of X,the carrier of Y
by A1, Def5;
end;

:: deftheorem Def7 defines modetrans RSSPACE4:def 7 :
for X being non empty set
for Y being RealNormSpace
for f being set st f in BoundedFunctions X,Y holds
modetrans f,X,Y = f;

definition
let X be non empty set ;
let Y be RealNormSpace;
let u be Function of X,the carrier of Y;
func PreNorms u -> non empty Subset of REAL equals :: RSSPACE4:def 8
{ ||.(u . t).|| where t is Element of X : verum } ;
coherence
{ ||.(u . t).|| where t is Element of X : verum } is non empty Subset of REAL
proof end;
end;

:: deftheorem defines PreNorms RSSPACE4:def 8 :
for X being non empty set
for Y being RealNormSpace
for u being Function of X,the carrier of Y holds PreNorms u = { ||.(u . t).|| where t is Element of X : verum } ;

theorem Th13: :: RSSPACE4:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for g being bounded Function of X,the carrier of Y holds
( not PreNorms g is empty & PreNorms g is bounded_above )
proof end;

theorem :: RSSPACE4:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for g being Function of X,the carrier of Y holds
( g is bounded iff PreNorms g is bounded_above )
proof end;

theorem Th15: :: RSSPACE4:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace ex NORM being Function of BoundedFunctions X,Y, REAL st
for f being set st f in BoundedFunctions X,Y holds
NORM . f = sup (PreNorms (modetrans f,X,Y))
proof end;

definition
let X be non empty set ;
let Y be RealNormSpace;
func BoundedFunctionsNorm X,Y -> Function of BoundedFunctions X,Y, REAL means :Def9: :: RSSPACE4:def 9
for x being set st x in BoundedFunctions X,Y holds
it . x = sup (PreNorms (modetrans x,X,Y));
existence
ex b1 being Function of BoundedFunctions X,Y, REAL st
for x being set st x in BoundedFunctions X,Y holds
b1 . x = sup (PreNorms (modetrans x,X,Y))
by Th15;
uniqueness
for b1, b2 being Function of BoundedFunctions X,Y, REAL st ( for x being set st x in BoundedFunctions X,Y holds
b1 . x = sup (PreNorms (modetrans x,X,Y)) ) & ( for x being set st x in BoundedFunctions X,Y holds
b2 . x = sup (PreNorms (modetrans x,X,Y)) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines BoundedFunctionsNorm RSSPACE4:def 9 :
for X being non empty set
for Y being RealNormSpace
for b3 being Function of BoundedFunctions X,Y, REAL holds
( b3 = BoundedFunctionsNorm X,Y iff for x being set st x in BoundedFunctions X,Y holds
b3 . x = sup (PreNorms (modetrans x,X,Y)) );

theorem Th16: :: RSSPACE4:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f being bounded Function of X,the carrier of Y holds modetrans f,X,Y = f
proof end;

theorem Th17: :: RSSPACE4:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f being bounded Function of X,the carrier of Y holds (BoundedFunctionsNorm X,Y) . f = sup (PreNorms f)
proof end;

definition
let X be non empty set ;
let Y be RealNormSpace;
func R_NormSpace_of_BoundedFunctions X,Y -> non empty NORMSTR equals :: RSSPACE4:def 10
NORMSTR(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(BoundedFunctionsNorm X,Y) #);
coherence
NORMSTR(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(BoundedFunctionsNorm X,Y) #) is non empty NORMSTR
;
end;

:: deftheorem defines R_NormSpace_of_BoundedFunctions RSSPACE4:def 10 :
for X being non empty set
for Y being RealNormSpace holds R_NormSpace_of_BoundedFunctions X,Y = NORMSTR(# (BoundedFunctions X,Y),(Zero_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Add_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(Mult_ (BoundedFunctions X,Y),(RealVectSpace X,Y)),(BoundedFunctionsNorm X,Y) #);

theorem Th18: :: RSSPACE4:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace holds X --> (0. Y) = 0. (R_NormSpace_of_BoundedFunctions X,Y)
proof end;

theorem Th19: :: RSSPACE4:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedFunctions X,Y)
for g being bounded Function of X,the carrier of Y st g = f holds
for t being Element of X holds ||.(g . t).|| <= ||.f.||
proof end;

theorem :: RSSPACE4:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedFunctions X,Y) holds 0 <= ||.f.||
proof end;

theorem Th21: :: RSSPACE4:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedFunctions X,Y) st f = 0. (R_NormSpace_of_BoundedFunctions X,Y) holds
0 = ||.f.||
proof end;

theorem Th22: :: RSSPACE4:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f, g, h being Point of (R_NormSpace_of_BoundedFunctions X,Y)
for f', g', h' being bounded Function of X,the carrier of Y st f' = f & g' = g & h' = h holds
( h = f + g iff for x being Element of X holds h' . x = (f' . x) + (g' . x) )
proof end;

theorem Th23: :: RSSPACE4:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f, h being Point of (R_NormSpace_of_BoundedFunctions X,Y)
for f', h' being bounded Function of X,the carrier of Y st f' = f & h' = h holds
for a being Real holds
( h = a * f iff for x being Element of X holds h' . x = a * (f' . x) )
proof end;

theorem Th24: :: RSSPACE4:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f, g being Point of (R_NormSpace_of_BoundedFunctions X,Y)
for a being Real holds
( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedFunctions X,Y) ) & ( f = 0. (R_NormSpace_of_BoundedFunctions X,Y) implies ||.f.|| = 0 ) & ||.(a * f).|| = (abs a) * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
proof end;

theorem Th25: :: RSSPACE4:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace holds R_NormSpace_of_BoundedFunctions X,Y is RealNormSpace-like
proof end;

theorem Th26: :: RSSPACE4:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace holds R_NormSpace_of_BoundedFunctions X,Y is RealNormSpace
proof end;

registration
let X be non empty set ;
let Y be RealNormSpace;
cluster R_NormSpace_of_BoundedFunctions X,Y -> non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like RealNormSpace-like ;
coherence
( R_NormSpace_of_BoundedFunctions X,Y is RealNormSpace-like & R_NormSpace_of_BoundedFunctions X,Y is RealLinearSpace-like & R_NormSpace_of_BoundedFunctions X,Y is Abelian & R_NormSpace_of_BoundedFunctions X,Y is add-associative & R_NormSpace_of_BoundedFunctions X,Y is right_zeroed & R_NormSpace_of_BoundedFunctions X,Y is right_complementable )
by Th26;
end;

theorem Th27: :: RSSPACE4:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace
for f, g, h being Point of (R_NormSpace_of_BoundedFunctions X,Y)
for f', g', h' being bounded Function of X,the carrier of Y st f' = f & g' = g & h' = h holds
( h = f - g iff for x being Element of X holds h' . x = (f' . x) - (g' . x) )
proof end;

Lm8: for e being Real
for seq being Real_Sequence st seq is convergent & ex k being Nat st
for i being Nat st k <= i holds
seq . i <= e holds
lim seq <= e
proof end;

theorem Th28: :: RSSPACE4:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealNormSpace st Y is complete holds
for seq being sequence of (R_NormSpace_of_BoundedFunctions X,Y) st seq is CCauchy holds
seq is convergent
proof end;

theorem Th29: :: RSSPACE4:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being non empty set
for Y being RealBanachSpace holds R_NormSpace_of_BoundedFunctions X,Y is RealBanachSpace
proof end;

registration
let X be non empty set ;
let Y be RealBanachSpace;
cluster R_NormSpace_of_BoundedFunctions X,Y -> non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like RealNormSpace-like complete ;
coherence
R_NormSpace_of_BoundedFunctions X,Y is complete
by Th29;
end;