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

0 = ||.f.||

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

assume A1: f = 0. (DualSp X) ; :: thesis: 0 = ||.f.||

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

0 = ||.f.||

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

assume A1: f = 0. (DualSp X) ; :: thesis: 0 = ||.f.||

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

proof

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

set z = the carrier of X --> 0;

reconsider z = the carrier of X --> 0 as Function of the carrier of X,REAL by FUNCOP_1:45, XREAL_0:def 1;

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;

A5: z = g by A1, Th31;

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

hence ||.f.|| = 0 by Th30; :: thesis: verum

end;set z = the carrier of X --> 0;

reconsider z = the carrier of X --> 0 as Function of the carrier of X,REAL by FUNCOP_1:45, XREAL_0:def 1;

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;

A5: z = g by A1, Th31;

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 ex t being VECTOR of X st

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

hence ( 0 <= r & r <= 0 ) by A5, COMPLEX1:44; :: thesis: verum

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

then ex t being VECTOR of X st

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

hence ( 0 <= r & r <= 0 ) by A5, COMPLEX1:44; :: thesis: verum

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

hence ||.f.|| = 0 by Th30; :: thesis: verum