let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f, g being Point of
for c being Complex holds
( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedFunctions X,Y) ) & ( f = 0. (C_NormSpace_of_BoundedFunctions X,Y) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
let Y be ComplexNormSpace; :: thesis: for f, g being Point of
for c being Complex holds
( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedFunctions X,Y) ) & ( f = 0. (C_NormSpace_of_BoundedFunctions X,Y) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
let f, g be Point of ; :: thesis: for c being Complex holds
( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedFunctions X,Y) ) & ( f = 0. (C_NormSpace_of_BoundedFunctions X,Y) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
let c be Complex; :: thesis: ( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedFunctions X,Y) ) & ( f = 0. (C_NormSpace_of_BoundedFunctions X,Y) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
A1: now
assume A2: f = 0. (C_NormSpace_of_BoundedFunctions X,Y) ; :: thesis: ||.f.|| = 0
thus ||.f.|| = 0 :: thesis: verum
proof
reconsider g = f as bounded Function of X,the carrier of Y by Def5;
set z = X --> (0. Y);
reconsider z = X --> (0. Y) as Function of X,the carrier of Y ;
consider r0 being set such that
A3: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A3;
A4: ( ( for s being real number st s in PreNorms g holds
s <= 0 ) implies sup (PreNorms g) <= 0 ) by SEQ_4:62;
A5: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th14;
A6: z = g by A2, Th19;
A7: now
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 Element of X such that
A8: r = ||.(g . t).|| ;
||.(g . t).|| = ||.(0. Y).|| by A6, FUNCOP_1:13
.= 0 by CLVECT_1:def 11 ;
hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum
end;
then 0 <= r0 by A3;
then sup (PreNorms g) = 0 by A7, A5, A3, A4, SEQ_4:def 4;
then (ComplexBoundedFunctionsNorm X,Y) . f = 0 by Th18;
hence ||.f.|| = 0 by CLVECT_1:def 10; :: thesis: verum
end;
end;
A9: ||.(f + g).|| <= ||.f.|| + ||.g.||
proof
reconsider f1 = f, g1 = g, h1 = f + g as bounded Function of X,the carrier of Y by Def5;
A10: ( ( for s being real number st s in PreNorms h1 holds
s <= ||.f.|| + ||.g.|| ) implies sup (PreNorms h1) <= ||.f.|| + ||.g.|| ) by SEQ_4:62;
A11: now
let t be Element of X; :: thesis: ||.(h1 . t).|| <= ||.f.|| + ||.g.||
( ||.(f1 . t).|| <= ||.f.|| & ||.(g1 . t).|| <= ||.g.|| ) by Th20;
then A12: ||.(f1 . t).|| + ||.(g1 . t).|| <= ||.f.|| + ||.g.|| by XREAL_1:9;
( ||.(h1 . t).|| = ||.((f1 . t) + (g1 . t)).|| & ||.((f1 . t) + (g1 . t)).|| <= ||.(f1 . t).|| + ||.(g1 . t).|| ) by Th23, CLVECT_1:def 11;
hence ||.(h1 . t).|| <= ||.f.|| + ||.g.|| by A12, XXREAL_0:2; :: thesis: verum
end;
A13: now
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 Element of X st r = ||.(h1 . t).|| ;
hence r <= ||.f.|| + ||.g.|| by A11; :: thesis: verum
end;
(ComplexBoundedFunctionsNorm X,Y) . (f + g) = sup (PreNorms h1) by Th18;
hence ||.(f + g).|| <= ||.f.|| + ||.g.|| by A13, A10, CLVECT_1:def 10; :: thesis: verum
end;
A14: ||.(c * f).|| = |.c.| * ||.f.||
proof
reconsider f1 = f, h1 = c * f as bounded Function of X,the carrier of Y by Def5;
A15: ( ( for s being real number st s in PreNorms h1 holds
s <= |.c.| * ||.f.|| ) implies sup (PreNorms h1) <= |.c.| * ||.f.|| ) by SEQ_4:62;
A16: now
let t be Element of X; :: thesis: ||.(h1 . t).|| <= |.c.| * ||.f.||
A17: 0 <= |.c.| by COMPLEX1:132;
( ||.(h1 . t).|| = ||.(c * (f1 . t)).|| & ||.(c * (f1 . t)).|| = |.c.| * ||.(f1 . t).|| ) by Th24, CLVECT_1:def 11;
hence ||.(h1 . t).|| <= |.c.| * ||.f.|| by A17, Th20, XREAL_1:66; :: thesis: verum
end;
A18: now
let r be Real; :: thesis: ( r in PreNorms h1 implies r <= |.c.| * ||.f.|| )
assume r in PreNorms h1 ; :: thesis: r <= |.c.| * ||.f.||
then ex t being Element of X st r = ||.(h1 . t).|| ;
hence r <= |.c.| * ||.f.|| by A16; :: thesis: verum
end;
A19: now
per cases ( c <> 0c or c = 0c ) ;
case A20: c <> 0c ; :: thesis: |.c.| * ||.f.|| <= ||.(c * f).||
A21: now
let t be Element of X; :: thesis: ||.(f1 . t).|| <= (|.c.| " ) * ||.(c * f).||
A22: |.(c " ).| = |.(1r / c).| by COMPLEX1:def 7, XCMPLX_1:217
.= 1 / |.c.| by COMPLEX1:134, COMPLEX1:153
.= 1 * (|.c.| " ) by XCMPLX_0:def 9
.= |.c.| " ;
h1 . t = c * (f1 . t) by Th24;
then A23: (c " ) * (h1 . t) = ((c " ) * c) * (f1 . t) by CLVECT_1:def 2
.= 1r * (f1 . t) by A20, COMPLEX1:def 7, XCMPLX_0:def 7
.= f1 . t by CLVECT_1:def 2 ;
( ||.((c " ) * (h1 . t)).|| = |.(c " ).| * ||.(h1 . t).|| & 0 <= |.(c " ).| ) by CLVECT_1:def 11, COMPLEX1:132;
hence ||.(f1 . t).|| <= (|.c.| " ) * ||.(c * f).|| by A23, A22, Th20, XREAL_1:66; :: thesis: verum
end;
A24: now
let r be Real; :: thesis: ( r in PreNorms f1 implies r <= (|.c.| " ) * ||.(c * f).|| )
assume r in PreNorms f1 ; :: thesis: r <= (|.c.| " ) * ||.(c * f).||
then ex t being Element of X st r = ||.(f1 . t).|| ;
hence r <= (|.c.| " ) * ||.(c * f).|| by A21; :: thesis: verum
end;
A25: ( ( for s being real number st s in PreNorms f1 holds
s <= (|.c.| " ) * ||.(c * f).|| ) implies sup (PreNorms f1) <= (|.c.| " ) * ||.(c * f).|| ) by SEQ_4:62;
A26: 0 <= |.c.| by COMPLEX1:132;
(ComplexBoundedFunctionsNorm X,Y) . f = sup (PreNorms f1) by Th18;
then ||.f.|| <= (|.c.| " ) * ||.(c * f).|| by A24, A25, CLVECT_1:def 10;
then |.c.| * ||.f.|| <= |.c.| * ((|.c.| " ) * ||.(c * f).||) by A26, XREAL_1:66;
then A27: |.c.| * ||.f.|| <= (|.c.| * (|.c.| " )) * ||.(c * f).|| ;
|.c.| <> 0 by A20, COMPLEX1:133;
then |.c.| * ||.f.|| <= 1 * ||.(c * f).|| by A27, XCMPLX_0:def 7;
hence |.c.| * ||.f.|| <= ||.(c * f).|| ; :: thesis: verum
end;
end;
end;
(ComplexBoundedFunctionsNorm X,Y) . (c * f) = sup (PreNorms h1) by Th18;
then ||.(c * f).|| <= |.c.| * ||.f.|| by A18, A15, CLVECT_1:def 10;
hence ||.(c * f).|| = |.c.| * ||.f.|| by A19, XXREAL_0:1; :: thesis: verum
end;
now
reconsider g = f as bounded Function of X,the carrier of Y by Def5;
set z = X --> (0. Y);
reconsider z = X --> (0. Y) as Function of X,the carrier of Y ;
assume A29: ||.f.|| = 0 ; :: thesis: f = 0. (C_NormSpace_of_BoundedFunctions X,Y)
now
let t be Element of X; :: thesis: g . t = z . t
||.(g . t).|| <= ||.f.|| by Th20;
then ||.(g . t).|| = 0 by A29, CLVECT_1:106;
hence g . t = 0. Y by CLVECT_1:def 11
.= z . t by FUNCOP_1:13 ;
:: thesis: verum
end;
then g = z by FUNCT_2:113;
hence f = 0. (C_NormSpace_of_BoundedFunctions X,Y) by Th19; :: thesis: verum
end;
hence ( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedFunctions X,Y) ) & ( f = 0. (C_NormSpace_of_BoundedFunctions X,Y) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| ) by A1, A14, A9; :: thesis: verum