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.|| )

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 )

A9:
||.(f + g).|| <= ||.f.|| + ||.g.||
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;

then upper_bound (PreNorms g) = 0 by A7, A3, A4, SEQ_4:def 1;

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

end;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 )

then
0 <= r0
by A3;( 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;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

then upper_bound (PreNorms g) = 0 by A7, A3, A4, SEQ_4:def 1;

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

proof

A19:
||.(a * f).|| = |.a.| * ||.f.||
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;

end;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.||

|.(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;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

now :: thesis: for r being Real st r in PreNorms h1 holds

r <= ||.f.|| + ||.g.||

hence
||.(f + g).|| <= ||.f.|| + ||.g.||
by A10, Th30; :: thesis: verumr <= ||.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;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

proof

reconsider f1 = f, h1 = a * f as Lipschitzian linear-Functional of X by Def9;

then ||.(a * f).|| <= |.a.| * ||.f.|| by A27, SEQ_4:45;

hence ||.(a * f).|| = |.a.| * ||.f.|| by A28, XXREAL_0:1; :: thesis: verum

end;A21: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds

|.(h1 . t).| <= |.a.| * ||.f.||

|.(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;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

A27: now :: thesis: for r being Real st r in PreNorms h1 holds

r <= |.a.| * ||.f.||

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;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

A28: now :: thesis: ( ( a <> 0 & |.a.| * ||.f.|| <= ||.(a * f).|| ) or ( a = 0 & |.a.| * ||.f.|| = ||.(a * f).|| ) )end;

(BoundedLinearFunctionalsNorm X) . (a * f) = upper_bound (PreNorms h1)
by Th30;per cases
( a <> 0 or a = 0 )
;

end;

case A29:
a <> 0
; :: thesis: |.a.| * ||.f.|| <= ||.(a * f).||

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;

A30: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds

|.(f1 . t).| <= (|.a.| ") * ||.(a * f).||

|.(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;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

A37: now :: thesis: for r being Real st r in PreNorms f1 holds

r <= (|.a.| ") * ||.(a * f).||

A38:
( ( for s being Real st s 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;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

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

case A41:
a = 0
; :: thesis: |.a.| * ||.f.|| = ||.(a * f).||

reconsider fz = f as VECTOR of (R_VectorSpace_of_BoundedLinearFunctionals X) ;

A42: a * f = a * fz

.= 0. (R_VectorSpace_of_BoundedLinearFunctionals X) by A41, RLVECT_1:10

.= 0. (DualSp X) ;

thus |.a.| * ||.f.|| = 0 * ||.f.|| by A41, ABSVALUE:2

.= ||.(a * f).|| by A42, Th34 ; :: thesis: verum

end;A42: a * f = a * fz

.= 0. (R_VectorSpace_of_BoundedLinearFunctionals X) by A41, RLVECT_1:10

.= 0. (DualSp X) ;

thus |.a.| * ||.f.|| = 0 * ||.f.|| by A41, ABSVALUE:2

.= ||.(a * f).|| by A42, Th34 ; :: thesis: verum

then ||.(a * f).|| <= |.a.| * ||.f.|| by A27, SEQ_4:45;

hence ||.(a * f).|| = |.a.| * ||.f.|| by A28, XXREAL_0:1; :: thesis: verum

now :: thesis: ( ||.f.|| = 0 implies f = 0. (DualSp X) )

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: verumreconsider 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)

hence f = 0. (DualSp X) by Th31; :: 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;

assume A43: ||.f.|| = 0 ; :: thesis: f = 0. (DualSp X)

now :: thesis: for t being VECTOR of X holds g . t = z . t

then
g = z
;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;|.(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

hence f = 0. (DualSp X) by Th31; :: thesis: verum