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

for f being Point of (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)) holds 0 <= ||.f.||

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

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

consider r0 being object such that

A1: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A1;

A2: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th12;

A3: (ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms g) by Th15;

then 0 <= upper_bound (PreNorms g) by A2, A1, SEQ_4:def 1;

hence 0 <= ||.f.|| by A3; :: thesis: verum

for f being Point of (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)) holds 0 <= ||.f.||

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

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

consider r0 being object such that

A1: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A1;

A2: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th12;

A3: (ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms g) by Th15;

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

0 <= r

then
0 <= r0
by A1;0 <= r

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

assume r in PreNorms g ; :: thesis: 0 <= r

then ex t being Element of X st r = ||.(g . t).|| ;

hence 0 <= r by CLVECT_1:105; :: thesis: verum

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

then ex t being Element of X st r = ||.(g . t).|| ;

hence 0 <= r by CLVECT_1:105; :: thesis: verum

then 0 <= upper_bound (PreNorms g) by A2, A1, SEQ_4:def 1;

hence 0 <= ||.f.|| by A3; :: thesis: verum