:: by Yasumasa Suzuki

::

:: Received January 6, 2004

:: Copyright (c) 2004-2021 Association of Mizar Users

Lm1: for rseq being Real_Sequence

for K being Real st ( for n being Nat holds rseq . n <= K ) holds

upper_bound (rng rseq) <= K

proof end;

Lm2: for rseq being Real_Sequence st rseq is bounded holds

for n being Nat holds rseq . n <= upper_bound (rng rseq)

proof end;

definition

ex b_{1} being Subset of Linear_Space_of_RealSequences st

for x being object holds

( x in b_{1} iff ( x in the_set_of_RealSequences & seq_id x is bounded ) )

for b_{1}, b_{2} being Subset of Linear_Space_of_RealSequences st ( for x being object holds

( x in b_{1} iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) ) & ( for x being object holds

( x in b_{2} iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) ) holds

b_{1} = b_{2}
end;

func the_set_of_BoundedRealSequences -> Subset of Linear_Space_of_RealSequences means :Def1: :: RSSPACE4:def 1

for x being object holds

( x in it iff ( x in the_set_of_RealSequences & seq_id x is bounded ) );

existence for x being object holds

( x in it iff ( x in the_set_of_RealSequences & seq_id x is bounded ) );

ex b

for x being object holds

( x in b

proof end;

uniqueness for b

( x in b

( x in b

b

proof end;

:: deftheorem Def1 defines the_set_of_BoundedRealSequences RSSPACE4:def 1 :

for b_{1} being Subset of Linear_Space_of_RealSequences holds

( b_{1} = the_set_of_BoundedRealSequences iff for x being object holds

( x in b_{1} iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) );

for b

( b

( x in b

registration

coherence

not the_set_of_BoundedRealSequences is empty

the_set_of_BoundedRealSequences is linearly-closed

end;
not the_set_of_BoundedRealSequences is empty

proof end;

coherence the_set_of_BoundedRealSequences is linearly-closed

proof 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:11;

registration

( 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 vector-distributive & 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 scalar-distributive & 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 scalar-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 scalar-unital ) by RSSPACE:11;

end;

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)) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

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 vector-distributive & 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 scalar-distributive & 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 scalar-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 scalar-unital ) by RSSPACE:11;

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 vector-distributive & 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 scalar-distributive & 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 scalar-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 scalar-unital )

;

definition

ex b_{1} being Function of the_set_of_BoundedRealSequences,REAL st

for x being object st x in the_set_of_BoundedRealSequences holds

b_{1} . x = upper_bound (rng (abs (seq_id x)))

for b_{1}, b_{2} being Function of the_set_of_BoundedRealSequences,REAL st ( for x being object st x in the_set_of_BoundedRealSequences holds

b_{1} . x = upper_bound (rng (abs (seq_id x))) ) & ( for x being object st x in the_set_of_BoundedRealSequences holds

b_{2} . x = upper_bound (rng (abs (seq_id x))) ) holds

b_{1} = b_{2}
end;

func linfty_norm -> Function of the_set_of_BoundedRealSequences,REAL means :Def2: :: RSSPACE4:def 2

for x being object st x in the_set_of_BoundedRealSequences holds

it . x = upper_bound (rng (abs (seq_id x)));

existence for x being object st x in the_set_of_BoundedRealSequences holds

it . x = upper_bound (rng (abs (seq_id x)));

ex b

for x being object st x in the_set_of_BoundedRealSequences holds

b

proof end;

uniqueness for b

b

b

b

proof end;

:: deftheorem Def2 defines linfty_norm RSSPACE4:def 2 :

for b_{1} being Function of the_set_of_BoundedRealSequences,REAL holds

( b_{1} = linfty_norm iff for x being object st x in the_set_of_BoundedRealSequences holds

b_{1} . x = upper_bound (rng (abs (seq_id x))) );

for b

( b

b

Lm5: for rseq being Real_Sequence st ( for n being Nat holds rseq . n = In (0,REAL) ) holds

( rseq is bounded & upper_bound (rng (abs rseq)) = 0 )

proof end;

Lm6: for rseq being Real_Sequence st rseq is bounded & upper_bound (rng (abs rseq)) = 0 holds

for n being Nat holds rseq . n = 0

proof end;

theorem :: RSSPACE4:1

registration

( 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 vector-distributive & 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 scalar-distributive & 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 scalar-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 scalar-unital ) by Lm4, RSSPACE3:2;

end;

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 #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

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 vector-distributive & 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 scalar-distributive & 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 scalar-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 scalar-unital ) by Lm4, RSSPACE3:2;

definition

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;

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 #);

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 ;

:: 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 #);

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 Th2: :: RSSPACE4:2

( 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.|| = upper_bound (rng (abs (seq_id v))) ) )

( 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.|| = upper_bound (rng (abs (seq_id v))) ) )

proof end;

theorem Th3: :: RSSPACE4:3

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).|| = |.a.| * ||.x.|| )

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).|| = |.a.| * ||.x.|| )

proof end;

registration

( linfty_Space is reflexive & linfty_Space is discerning & linfty_Space is RealNormSpace-like & linfty_Space is vector-distributive & linfty_Space is scalar-distributive & linfty_Space is scalar-associative & linfty_Space is scalar-unital & linfty_Space is Abelian & linfty_Space is add-associative & linfty_Space is right_zeroed & linfty_Space is right_complementable ) by Th3;

end;

cluster linfty_Space -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ;

coherence ( linfty_Space is reflexive & linfty_Space is discerning & linfty_Space is RealNormSpace-like & linfty_Space is vector-distributive & linfty_Space is scalar-distributive & linfty_Space is scalar-associative & linfty_Space is scalar-unital & linfty_Space is Abelian & linfty_Space is add-associative & linfty_Space is right_zeroed & linfty_Space is right_complementable ) by Th3;

definition
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 ) ) );

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 Th5: :: RSSPACE4:5

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

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;

ex b_{1} being Function of X, the carrier of Y st b_{1} is bounded

end;
let Y be RealNormSpace;

cluster Relation-like X -defined the carrier of Y -valued non empty Function-like V26(X) quasi_total bounded for Element of bool [:X, the carrier of Y:];

existence ex b

proof end;

definition

let X be non empty set ;

let Y be RealNormSpace;

ex b_{1} being Subset of (RealVectSpace (X,Y)) st

for x being set holds

( x in b_{1} iff x is bounded Function of X, the carrier of Y )

for b_{1}, b_{2} being Subset of (RealVectSpace (X,Y)) st ( for x being set holds

( x in b_{1} iff x is bounded Function of X, the carrier of Y ) ) & ( for x being set holds

( x in b_{2} iff x is bounded Function of X, the carrier of Y ) ) holds

b_{1} = b_{2}

end;
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 for x being set holds

( x in it iff x is bounded Function of X, the carrier of Y );

ex b

for x being set holds

( x in b

proof end;

uniqueness for b

( x in b

( x in b

b

proof end;

:: deftheorem Def5 defines BoundedFunctions RSSPACE4:def 5 :

for X being non empty set

for Y being RealNormSpace

for b_{3} being Subset of (RealVectSpace (X,Y)) holds

( b_{3} = BoundedFunctions (X,Y) iff for x being set holds

( x in b_{3} iff x is bounded Function of X, the carrier of Y ) );

for X being non empty set

for Y being RealNormSpace

for b

( b

( x in b

registration

let X be non empty set ;

let Y be RealNormSpace;

coherence

not BoundedFunctions (X,Y) is empty

end;
let Y be RealNormSpace;

coherence

not BoundedFunctions (X,Y) is empty

proof end;

theorem :: RSSPACE4:7

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) by Th6, RSSPACE:11;

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) by Th6, RSSPACE:11;

registration

let X be non empty set ;

let Y be RealNormSpace;

( 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 vector-distributive & 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 scalar-distributive & 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 scalar-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 scalar-unital ) by Th6, RSSPACE:11;

end;
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)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

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 vector-distributive & 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 scalar-distributive & 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 scalar-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 scalar-unital ) by Th6, RSSPACE:11;

definition

let X be non empty set ;

let Y be RealNormSpace;

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;
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)))) #);

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 ;

:: 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)))) #);

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;

coherence

R_VectorSpace_of_BoundedFunctions (X,Y) is strict ;

end;
let Y be RealNormSpace;

coherence

R_VectorSpace_of_BoundedFunctions (X,Y) is strict ;

theorem Th8: :: RSSPACE4:8

for X being non empty set

for Y being RealNormSpace

for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y))

for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds

( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

for Y being RealNormSpace

for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y))

for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds

( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

proof end;

theorem Th9: :: RSSPACE4:9

for X being non empty set

for Y being RealNormSpace

for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y))

for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds

for a being Real holds

( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )

for Y being RealNormSpace

for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y))

for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds

for a being Real holds

( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )

proof end;

theorem Th10: :: RSSPACE4:10

for X being non empty set

for Y being RealNormSpace holds 0. (R_VectorSpace_of_BoundedFunctions (X,Y)) = X --> (0. Y)

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 object ;

assume A1: f in BoundedFunctions (X,Y) ;

coherence

f is bounded Function of X, the carrier of Y by A1, Def5;

end;
let Y be RealNormSpace;

let f be object ;

assume A1: f in BoundedFunctions (X,Y) ;

coherence

f is bounded Function of X, the carrier of Y by A1, Def5;

:: deftheorem Def7 defines modetrans RSSPACE4:def 7 :

for X being non empty set

for Y being RealNormSpace

for f being object st f in BoundedFunctions (X,Y) holds

modetrans (f,X,Y) = f;

for X being non empty set

for Y being RealNormSpace

for f being object 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;

{ ||.(u . t).|| where t is Element of X : verum } is non empty Subset of REAL

end;
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 } ;

{ ||.(u . t).|| where t is Element of X : verum } is non empty Subset of REAL

proof 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 } ;

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 Th11: :: RSSPACE4:11

for X being non empty set

for Y being RealNormSpace

for g being bounded Function of X, the carrier of Y holds PreNorms g is bounded_above

for Y being RealNormSpace

for g being bounded Function of X, the carrier of Y holds PreNorms g is bounded_above

proof end;

theorem :: RSSPACE4:12

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 )

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;

definition

let X be non empty set ;

let Y be RealNormSpace;

ex b_{1} being Function of (BoundedFunctions (X,Y)),REAL st

for x being object st x in BoundedFunctions (X,Y) holds

b_{1} . x = upper_bound (PreNorms (modetrans (x,X,Y)))

for b_{1}, b_{2} being Function of (BoundedFunctions (X,Y)),REAL st ( for x being object st x in BoundedFunctions (X,Y) holds

b_{1} . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) & ( for x being object st x in BoundedFunctions (X,Y) holds

b_{2} . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) holds

b_{1} = b_{2}

end;
let Y be RealNormSpace;

func BoundedFunctionsNorm (X,Y) -> Function of (BoundedFunctions (X,Y)),REAL means :Def9: :: RSSPACE4:def 9

for x being object st x in BoundedFunctions (X,Y) holds

it . x = upper_bound (PreNorms (modetrans (x,X,Y)));

existence for x being object st x in BoundedFunctions (X,Y) holds

it . x = upper_bound (PreNorms (modetrans (x,X,Y)));

ex b

for x being object st x in BoundedFunctions (X,Y) holds

b

proof end;

uniqueness for b

b

b

b

proof end;

:: deftheorem Def9 defines BoundedFunctionsNorm RSSPACE4:def 9 :

for X being non empty set

for Y being RealNormSpace

for b_{3} being Function of (BoundedFunctions (X,Y)),REAL holds

( b_{3} = BoundedFunctionsNorm (X,Y) iff for x being object st x in BoundedFunctions (X,Y) holds

b_{3} . x = upper_bound (PreNorms (modetrans (x,X,Y))) );

for X being non empty set

for Y being RealNormSpace

for b

( b

b

theorem Th13: :: RSSPACE4:13

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

for Y being RealNormSpace

for f being bounded Function of X, the carrier of Y holds modetrans (f,X,Y) = f

proof end;

theorem Th14: :: RSSPACE4:14

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 = upper_bound (PreNorms f)

for Y being RealNormSpace

for f being bounded Function of X, the carrier of Y holds (BoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f)

proof end;

definition

let X be non empty set ;

let Y be RealNormSpace;

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;
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)) #);

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 ;

:: 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)) #);

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 Th15: :: RSSPACE4:15

for X being non empty set

for Y being RealNormSpace holds X --> (0. Y) = 0. (R_NormSpace_of_BoundedFunctions (X,Y))

for Y being RealNormSpace holds X --> (0. Y) = 0. (R_NormSpace_of_BoundedFunctions (X,Y))

proof end;

theorem Th16: :: RSSPACE4:16

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.||

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 :: RSSPACE4:17

for X being non empty set

for Y being RealNormSpace

for f being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) holds 0 <= ||.f.||

for Y being RealNormSpace

for f being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) holds 0 <= ||.f.||

proof end;

theorem Th18: :: RSSPACE4:18

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.||

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 Th19: :: RSSPACE4:19

for X being non empty set

for Y being RealNormSpace

for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))

for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds

( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

for Y being RealNormSpace

for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))

for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds

( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

proof end;

theorem Th20: :: RSSPACE4:20

for X being non empty set

for Y being RealNormSpace

for f, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))

for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds

for a being Real holds

( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )

for Y being RealNormSpace

for f, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))

for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds

for a being Real holds

( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) )

proof end;

theorem Th21: :: RSSPACE4:21

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).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

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).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

proof end;

theorem Th22: :: RSSPACE4:22

for X being non empty set

for Y being RealNormSpace holds

( R_NormSpace_of_BoundedFunctions (X,Y) is reflexive & R_NormSpace_of_BoundedFunctions (X,Y) is discerning & R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace-like ) by Th21;

for Y being RealNormSpace holds

( R_NormSpace_of_BoundedFunctions (X,Y) is reflexive & R_NormSpace_of_BoundedFunctions (X,Y) is discerning & R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace-like ) by Th21;

theorem Th23: :: RSSPACE4:23

for X being non empty set

for Y being RealNormSpace holds R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace

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;

( R_NormSpace_of_BoundedFunctions (X,Y) is reflexive & R_NormSpace_of_BoundedFunctions (X,Y) is discerning & R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace-like & R_NormSpace_of_BoundedFunctions (X,Y) is vector-distributive & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-distributive & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-associative & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-unital & 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 Th23;

end;
let Y be RealNormSpace;

cluster R_NormSpace_of_BoundedFunctions (X,Y) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ;

coherence ( R_NormSpace_of_BoundedFunctions (X,Y) is reflexive & R_NormSpace_of_BoundedFunctions (X,Y) is discerning & R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace-like & R_NormSpace_of_BoundedFunctions (X,Y) is vector-distributive & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-distributive & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-associative & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-unital & 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 Th23;

theorem Th24: :: RSSPACE4:24

for X being non empty set

for Y being RealNormSpace

for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))

for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds

( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )

for Y being RealNormSpace

for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))

for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds

( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )

proof end;

Lm7: 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 Th25: :: RSSPACE4:25

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 Cauchy_sequence_by_Norm holds

seq is convergent

for Y being RealNormSpace st Y is complete holds

for seq being sequence of (R_NormSpace_of_BoundedFunctions (X,Y)) st seq is Cauchy_sequence_by_Norm holds

seq is convergent

proof end;

theorem Th26: :: RSSPACE4:26

for X being non empty set

for Y being RealBanachSpace holds R_NormSpace_of_BoundedFunctions (X,Y) is RealBanachSpace

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;

coherence

R_NormSpace_of_BoundedFunctions (X,Y) is complete by Th26;

end;
let Y be RealBanachSpace;

coherence

R_NormSpace_of_BoundedFunctions (X,Y) is complete by Th26;