let X be RealNormSpace; :: thesis: for g being linear-Functional of X holds
( g is Lipschitzian iff PreNorms g is bounded_above )
let g be linear-Functional of X; :: thesis: ( g is Lipschitzian iff PreNorms g is bounded_above )

now :: thesis: ( PreNorms g is bounded_above implies ex K being Real st g is Lipschitzian )

reconsider K = upper_bound (PreNorms g) as Real ;
assume A1: PreNorms g is bounded_above ; :: thesis: ex K being Real st g is Lipschitzian

A2: now :: thesis: for t being VECTOR of X holds |.(g . t).| <= K * ||.t.||

let t be VECTOR of X; :: thesis: |.(g . b₁).| <= K * ||.b₁.||

per cases ( t = 0. X or t <> 0. X ) ;

suppose A3: t = 0. X ; :: thesis: |.(g . b₁).| <= K * ||.b₁.||

then A4: ||.t.|| = 0 ;
g . t = g . (0 * (0. X)) by A3
.= 0 * (g . (0. X)) by HAHNBAN:def 3
.= 0 ;
hence |.(g . t).| <= K * ||.t.|| by A4, COMPLEX1:44; :: thesis: verum

end;

suppose A5: t <> 0. X ; :: thesis: |.(g . b₁).| <= K * ||.b₁.||

reconsider t1 = (||.t.|| ") * t as VECTOR of X ;
A6: ||.t.|| <> 0 by A5, NORMSP_0:def 5;
A7: (|.(g . t).| / ||.t.||) * ||.t.|| = (|.(g . t).| * (||.t.|| ")) * ||.t.|| by XCMPLX_0:def 9
.= |.(g . t).| * ((||.t.|| ") * ||.t.||)
.= |.(g . t).| * 1 by A6, XCMPLX_0:def 7
.= |.(g . t).| ;
A8: |.(||.t.|| ").| = |.(1 * (||.t.|| ")).|
.= |.(1 / ||.t.||).| by XCMPLX_0:def 9
.= 1 / ||.t.|| by ABSVALUE:def 1
.= 1 * (||.t.|| ") by XCMPLX_0:def 9
.= ||.t.|| " ;
||.t1.|| = |.(||.t.|| ").| * ||.t.|| by NORMSP_1:def 1
.= 1 by A6, A8, XCMPLX_0:def 7 ;
then A9: |.(g . t1).| in { |.(g . s).| where s is VECTOR of X : ||.s.|| <= 1 } ;
|.(g . t).| / ||.t.|| = |.(g . t).| * (||.t.|| ") by XCMPLX_0:def 9
.= |.((||.t.|| ") * (g . t)).| by A8, COMPLEX1:65
.= |.(g . t1).| by HAHNBAN:def 3 ;
then |.(g . t).| / ||.t.|| <= K by A1, A9, SEQ_4:def 1;
hence |.(g . t).| <= K * ||.t.|| by A7, XREAL_1:64; :: thesis: verum

end;

end;

end;

take K = K; :: thesis: g is Lipschitzian
0 <= K

proof

consider r0 being object such that
A10: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A10;

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

then 0 <= r0 by A10;
hence 0 <= K by A1, A10, SEQ_4:def 1; :: thesis: verum

end;

hence g is Lipschitzian by A2; :: thesis: verum

end;

hence ( g is Lipschitzian iff PreNorms g is bounded_above ) ; :: thesis: verum