let X be non empty set ; :: thesis: for Y being ComplexNormSpace

for f being Point of (C_NormSpace_of_BoundedFunctions (X,Y)) st f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) holds

0 = ||.f.||

let Y be ComplexNormSpace; :: thesis: for f being Point of (C_NormSpace_of_BoundedFunctions (X,Y)) st f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) holds

0 = ||.f.||

let f be Point of (C_NormSpace_of_BoundedFunctions (X,Y)); :: thesis: ( f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) implies 0 = ||.f.|| )

assume A1: f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) ; :: thesis: 0 = ||.f.||

thus ||.f.|| = 0 :: thesis: verum

for f being Point of (C_NormSpace_of_BoundedFunctions (X,Y)) st f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) holds

0 = ||.f.||

let Y be ComplexNormSpace; :: thesis: for f being Point of (C_NormSpace_of_BoundedFunctions (X,Y)) st f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) holds

0 = ||.f.||

let f be Point of (C_NormSpace_of_BoundedFunctions (X,Y)); :: thesis: ( f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) implies 0 = ||.f.|| )

assume A1: f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) ; :: thesis: 0 = ||.f.||

thus ||.f.|| = 0 :: thesis: verum

proof

reconsider g = f as bounded Function of X, the carrier of Y by Def5;

set z = X --> (0. Y);

reconsider z = X --> (0. Y) as Function of X, the carrier of Y ;

consider r0 being object such that

A2: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A2;

A3: ( ( for s being Real st s in PreNorms g holds

s <= 0 ) implies upper_bound (PreNorms g) <= 0 ) by SEQ_4:45;

A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th12;

A5: z = g by A1, Th16;

then upper_bound (PreNorms g) = 0 by A6, A4, A2, A3, SEQ_4:def 1;

then (ComplexBoundedFunctionsNorm (X,Y)) . f = 0 by Th15;

hence ||.f.|| = 0 ; :: thesis: verum

end;set z = X --> (0. Y);

reconsider z = X --> (0. Y) as Function of X, the carrier of Y ;

consider r0 being object such that

A2: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A2;

A3: ( ( for s being Real st s in PreNorms g holds

s <= 0 ) implies upper_bound (PreNorms g) <= 0 ) by SEQ_4:45;

A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th12;

A5: z = g by A1, Th16;

A6: now :: thesis: for r being Real st r in PreNorms g holds

( 0 <= r & r <= 0 )

then
0 <= r0
by A2;( 0 <= r & r <= 0 )

let r be Real; :: thesis: ( r in PreNorms g implies ( 0 <= r & r <= 0 ) )

assume r in PreNorms g ; :: thesis: ( 0 <= r & r <= 0 )

then consider t being Element of X such that

A7: r = ||.(g . t).|| ;

||.(g . t).|| = ||.(0. Y).|| by A5, FUNCOP_1:7

.= 0 ;

hence ( 0 <= r & r <= 0 ) by A7; :: thesis: verum

end;assume r in PreNorms g ; :: thesis: ( 0 <= r & r <= 0 )

then consider t being Element of X such that

A7: r = ||.(g . t).|| ;

||.(g . t).|| = ||.(0. Y).|| by A5, FUNCOP_1:7

.= 0 ;

hence ( 0 <= r & r <= 0 ) by A7; :: thesis: verum

then upper_bound (PreNorms g) = 0 by A6, A4, A2, A3, SEQ_4:def 1;

then (ComplexBoundedFunctionsNorm (X,Y)) . f = 0 by Th15;

hence ||.f.|| = 0 ; :: thesis: verum