let X be RealNormSpace; :: thesis: for f being Point of (DualSp X) holds 0 <= ||.f.||

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

reconsider g = f as Lipschitzian linear-Functional of X by Def9;

consider r0 being object such that

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

reconsider r0 = r0 as Real by A1;

A3: (BoundedLinearFunctionalsNorm X) . f = upper_bound (PreNorms g) by Th30;

hence 0 <= ||.f.|| by A1, SEQ_4:def 1, A3; :: thesis: verum

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

reconsider g = f as Lipschitzian linear-Functional of X by Def9;

consider r0 being object such that

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

reconsider r0 = r0 as Real by A1;

A3: (BoundedLinearFunctionalsNorm X) . f = upper_bound (PreNorms g) by Th30;

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 VECTOR of X st

( r = |.(g . t).| & ||.t.|| <= 1 ) ;

hence 0 <= r by COMPLEX1:46; :: thesis: verum

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

then ex t being VECTOR of X st

( r = |.(g . t).| & ||.t.|| <= 1 ) ;

hence 0 <= r by COMPLEX1:46; :: thesis: verum

hence 0 <= ||.f.|| by A1, SEQ_4:def 1, A3; :: thesis: verum