let X be RealNormSpace; :: thesis: for f, g being Point of (DualSp X)
for a being Real holds
( ( ||.f.|| = 0 implies f = 0. (DualSp X) ) & ( f = 0. (DualSp X) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
let f, g be Point of (DualSp X); :: thesis: for a being Real holds
( ( ||.f.|| = 0 implies f = 0. (DualSp X) ) & ( f = 0. (DualSp X) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
let a be Real; :: thesis: ( ( ||.f.|| = 0 implies f = 0. (DualSp X) ) & ( f = 0. (DualSp X) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
A1: now :: thesis: ( f = 0. (DualSp X) implies ||.f.|| = 0 )
assume A2: f = 0. (DualSp X) ; :: thesis: ||.f.|| = 0
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
A3: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A3;
A4: ( ( for s being Real st s in PreNorms g holds
s <= 0 ) implies upper_bound (PreNorms g) <= 0 ) by SEQ_4:45;
A6: z = g by A2, Th31;
A7: now :: thesis: for r being Real st r in PreNorms g holds
( 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 VECTOR of X such that
A8: r = |.(g . t).| and
||.t.|| <= 1 ;
thus ( 0 <= r & r <= 0 ) by A8, A6, COMPLEX1:44; :: thesis: verum
end;
then 0 <= r0 by A3;
then upper_bound (PreNorms g) = 0 by A7, A3, A4, SEQ_4:def 1;
hence ||.f.|| = 0 by Th30; :: thesis: verum
end;
A9: ||.(f + g).|| <= ||.f.|| + ||.g.||
proof
reconsider f1 = f, g1 = g, h1 = f + g as Lipschitzian linear-Functional of X by Def9;
A10: ( ( for s being Real st s in PreNorms h1 holds
s <= ||.f.|| + ||.g.|| ) implies upper_bound (PreNorms h1) <= ||.f.|| + ||.g.|| ) by SEQ_4:45;
A11: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds
|.(h1 . t).| <= ||.f.|| + ||.g.||
let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies |.(h1 . t).| <= ||.f.|| + ||.g.|| )
assume A12: ||.t.|| <= 1 ; :: thesis: |.(h1 . t).| <= ||.f.|| + ||.g.||
( 0 <= ||.f.|| & 0 <= ||.g.|| ) by Th33;
then ( ||.f.|| * ||.t.|| <= ||.f.|| * 1 & ||.g.|| * ||.t.|| <= ||.g.|| * 1 ) by A12, XREAL_1:64;
then A14: (||.f.|| * ||.t.||) + (||.g.|| * ||.t.||) <= (||.f.|| * 1) + (||.g.|| * 1) by XREAL_1:7;
A15: |.((f1 . t) + (g1 . t)).| <= |.(f1 . t).| + |.(g1 . t).| by COMPLEX1:56;
( |.(g1 . t).| <= ||.g.|| * ||.t.|| & |.(f1 . t).| <= ||.f.|| * ||.t.|| ) by Th32;
then |.(f1 . t).| + |.(g1 . t).| <= (||.f.|| * ||.t.||) + (||.g.|| * ||.t.||) by XREAL_1:7;
then A17: |.(f1 . t).| + |.(g1 . t).| <= ||.f.|| + ||.g.|| by A14, XXREAL_0:2;
|.(h1 . t).| = |.((f1 . t) + (g1 . t)).| by Th35;
hence |.(h1 . t).| <= ||.f.|| + ||.g.|| by A15, A17, XXREAL_0:2; :: thesis: verum
end;
now :: thesis: for r being Real st r in PreNorms h1 holds
r <= ||.f.|| + ||.g.||
let r be Real; :: thesis: ( r in PreNorms h1 implies r <= ||.f.|| + ||.g.|| )
assume r in PreNorms h1 ; :: thesis: r <= ||.f.|| + ||.g.||
then ex t being VECTOR of X st
( r = |.(h1 . t).| & ||.t.|| <= 1 ) ;
hence r <= ||.f.|| + ||.g.|| by A11; :: thesis: verum
end;
hence ||.(f + g).|| <= ||.f.|| + ||.g.|| by A10, Th30; :: thesis: verum
end;
A19: ||.(a * f).|| = |.a.| * ||.f.||
proof
reconsider f1 = f, h1 = a * f as Lipschitzian linear-Functional of X by Def9;
A21: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds
|.(h1 . t).| <= |.a.| * ||.f.||
A22: 0 <= ||.f.|| by Th33;
let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies |.(h1 . t).| <= |.a.| * ||.f.|| )
assume ||.t.|| <= 1 ; :: thesis: |.(h1 . t).| <= |.a.| * ||.f.||
then A23: ||.f.|| * ||.t.|| <= ||.f.|| * 1 by A22, XREAL_1:64;
|.(f1 . t).| <= ||.f.|| * ||.t.|| by Th32;
then A24: |.(f1 . t).| <= ||.f.|| by A23, XXREAL_0:2;
A25: |.(a * (f1 . t)).| = |.a.| * |.(f1 . t).| by COMPLEX1:65;
A26: 0 <= |.a.| by COMPLEX1:46;
|.(h1 . t).| = |.(a * (f1 . t)).| by Th36;
hence |.(h1 . t).| <= |.a.| * ||.f.|| by A25, A24, A26, XREAL_1:64; :: thesis: verum
end;
A27: now :: thesis: for r being Real st r in PreNorms h1 holds
r <= |.a.| * ||.f.||
let r be Real; :: thesis: ( r in PreNorms h1 implies r <= |.a.| * ||.f.|| )
assume r in PreNorms h1 ; :: thesis: r <= |.a.| * ||.f.||
then ex t being VECTOR of X st
( r = |.(h1 . t).| & ||.t.|| <= 1 ) ;
hence r <= |.a.| * ||.f.|| by A21; :: thesis: verum
end;
A28: now :: thesis: ( ( a <> 0 & |.a.| * ||.f.|| <= ||.(a * f).|| ) or ( a = 0 & |.a.| * ||.f.|| = ||.(a * f).|| ) )
per cases ( a <> 0 or a = 0 ) ;
case A29: a <> 0 ; :: thesis: |.a.| * ||.f.|| <= ||.(a * f).||
A30: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds
|.(f1 . t).| <= (|.a.| ") * ||.(a * f).||
A31: 0 <= ||.(a * f).|| by Th33;
let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies |.(f1 . t).| <= (|.a.| ") * ||.(a * f).|| )
assume ||.t.|| <= 1 ; :: thesis: |.(f1 . t).| <= (|.a.| ") * ||.(a * f).||
then A32: ||.(a * f).|| * ||.t.|| <= ||.(a * f).|| * 1 by A31, XREAL_1:64;
|.(h1 . t).| <= ||.(a * f).|| * ||.t.|| by Th32;
then A33: |.(h1 . t).| <= ||.(a * f).|| by A32, XXREAL_0:2;
h1 . t = a * (f1 . t) by Th36;
then A34: (a ") * (h1 . t) = ((a ") * a) * (f1 . t)
.= 1 * (f1 . t) by A29, XCMPLX_0:def 7
.= f1 . t ;
A35: |.(a ").| = |.(1 * (a ")).|
.= |.(1 / a).| by XCMPLX_0:def 9
.= 1 / |.a.| by ABSVALUE:7
.= 1 * (|.a.| ") by XCMPLX_0:def 9
.= |.a.| " ;
A36: 0 <= |.(a ").| by COMPLEX1:46;
|.((a ") * (h1 . t)).| = |.(a ").| * |.(h1 . t).| by COMPLEX1:65;
hence |.(f1 . t).| <= (|.a.| ") * ||.(a * f).|| by A34, A33, A36, A35, XREAL_1:64; :: thesis: verum
end;
A37: now :: thesis: for r being Real st r in PreNorms f1 holds
r <= (|.a.| ") * ||.(a * f).||
let r be Real; :: thesis: ( r in PreNorms f1 implies r <= (|.a.| ") * ||.(a * f).|| )
assume r in PreNorms f1 ; :: thesis: r <= (|.a.| ") * ||.(a * f).||
then ex t being VECTOR of X st
( r = |.(f1 . t).| & ||.t.|| <= 1 ) ;
hence r <= (|.a.| ") * ||.(a * f).|| by A30; :: thesis: verum
end;
A38: ( ( for s being Real st s in PreNorms f1 holds
s <= (|.a.| ") * ||.(a * f).|| ) implies upper_bound (PreNorms f1) <= (|.a.| ") * ||.(a * f).|| ) by SEQ_4:45;
A39: 0 <= |.a.| by COMPLEX1:46;
||.f.|| <= (|.a.| ") * ||.(a * f).|| by A37, A38, Th30;
then |.a.| * ||.f.|| <= |.a.| * ((|.a.| ") * ||.(a * f).||) by A39, XREAL_1:64;
then A40: |.a.| * ||.f.|| <= (|.a.| * (|.a.| ")) * ||.(a * f).|| ;
|.a.| <> 0 by A29, COMPLEX1:47;
then |.a.| * ||.f.|| <= 1 * ||.(a * f).|| by A40, XCMPLX_0:def 7;
hence |.a.| * ||.f.|| <= ||.(a * f).|| ; :: thesis: verum
end;
end;
end;
(BoundedLinearFunctionalsNorm X) . (a * f) = upper_bound (PreNorms h1) by Th30;
then ||.(a * f).|| <= |.a.| * ||.f.|| by A27, SEQ_4:45;
hence ||.(a * f).|| = |.a.| * ||.f.|| by A28, XXREAL_0:1; :: thesis: verum
end;
now :: thesis: ( ||.f.|| = 0 implies f = 0. (DualSp X) )
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;
assume A43: ||.f.|| = 0 ; :: thesis: f = 0. (DualSp X)
now :: thesis: for t being VECTOR of X holds g . t = z . t
let t be VECTOR of X; :: thesis: g . t = z . t
|.(g . t).| <= ||.f.|| * ||.t.|| by Th32;
then |.(g . t).| = 0 by A43, COMPLEX1:46;
hence g . t = 0 by COMPLEX1:45
.= z . t ;
:: thesis: verum
end;
then g = z ;
hence f = 0. (DualSp X) by Th31; :: thesis: verum
end;
hence ( ( ||.f.|| = 0 implies f = 0. (DualSp X) ) & ( f = 0. (DualSp X) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| ) by A1, A19, A9; :: thesis: verum