let X, Y, Z be RealNormSpace; :: thesis: for f, g being Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z))
for a being Real holds
( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ) & ( f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
let f, g be Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)); :: thesis: for a being Real holds
( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ) & ( f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
let a be Real; :: thesis: ( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ) & ( f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
A1: now :: thesis: ( f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) implies ||.f.|| = 0 )
assume A2: f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ; :: thesis: ||.f.|| = 0
thus ||.f.|| = 0 :: thesis: verum
proof
reconsider g = f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
set z = the carrier of [:X,Y:] --> (0. Z);
reconsider z = the carrier of [:X,Y:] --> (0. Z) as Function of the carrier of [:X,Y:], the carrier of Z ;
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, s being VECTOR of Y such that
A8: r = ||.(g . (t,s)).|| and
( ||.t.|| <= 1 & ||.s.|| <= 1 ) ;
[t,s] is Point of [:X,Y:] ;
then g . (t,s) = 0. Z by A6, FUNCOP_1:7;
hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum
end;
then 0 <= r0 by A3;
then upper_bound (PreNorms g) = 0 by A3, A4, A7, SEQ_4:def 1;
hence ||.f.|| = 0 by Th30; :: thesis: verum
end;
end;
A9: ||.(f + g).|| <= ||.f.|| + ||.g.||
proof
reconsider f1 = f, g1 = g, h1 = f + g as Lipschitzian BilinearOperator of X,Y,Z 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
for s being VECTOR of Y st ||.t.|| <= 1 & ||.s.|| <= 1 holds
||.(h1 . (t,s)).|| <= ||.f.|| + ||.g.||
let t be VECTOR of X; :: thesis: for s being VECTOR of Y st ||.t.|| <= 1 & ||.s.|| <= 1 holds
||.(h1 . (t,s)).|| <= ||.f.|| + ||.g.||
let s be VECTOR of Y; :: thesis: ( ||.t.|| <= 1 & ||.s.|| <= 1 implies ||.(h1 . (t,s)).|| <= ||.f.|| + ||.g.|| )
assume A12: ( ||.t.|| <= 1 & ||.s.|| <= 1 ) ; :: thesis: ||.(h1 . (t,s)).|| <= ||.f.|| + ||.g.||
A13: ||.t.|| * ||.s.|| <= 1 * 1 by A12, XREAL_1:66;
0 <= ||.g.|| by Th33;
then A14: ||.g.|| * (||.t.|| * ||.s.||) <= ||.g.|| * 1 by A13, XREAL_1:64;
0 <= ||.f.|| by Th33;
then ||.f.|| * (||.t.|| * ||.s.||) <= ||.f.|| * 1 by A13, XREAL_1:64;
then A15: ((||.f.|| * ||.t.||) * ||.s.||) + ((||.g.|| * ||.t.||) * ||.s.||) <= (||.f.|| * 1) + (||.g.|| * 1) by A14, XREAL_1:7;
A16: ||.((f1 . (t,s)) + (g1 . (t,s))).|| <= ||.(f1 . (t,s)).|| + ||.(g1 . (t,s)).|| by NORMSP_1:def 1;
A17: ||.(g1 . (t,s)).|| <= (||.g.|| * ||.t.||) * ||.s.|| by Th32;
||.(f1 . (t,s)).|| <= (||.f.|| * ||.t.||) * ||.s.|| by Th32;
then ||.(f1 . (t,s)).|| + ||.(g1 . (t,s)).|| <= ((||.f.|| * ||.t.||) * ||.s.||) + ((||.g.|| * ||.t.||) * ||.s.||) by A17, XREAL_1:7;
then A18: ||.(f1 . (t,s)).|| + ||.(g1 . (t,s)).|| <= ||.f.|| + ||.g.|| by A15, XXREAL_0:2;
||.(h1 . (t,s)).|| = ||.((f1 . (t,s)) + (g1 . (t,s))).|| by Th35;
hence ||.(h1 . (t,s)).|| <= ||.f.|| + ||.g.|| by A16, A18, 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 ex s being VECTOR of Y st
( r = ||.(h1 . (t,s)).|| & ||.t.|| <= 1 & ||.s.|| <= 1 ) ;
hence r <= ||.f.|| + ||.g.|| by A11; :: thesis: verum
end;
hence ||.(f + g).|| <= ||.f.|| + ||.g.|| by A10, Th30; :: thesis: verum
end;
A20: ||.(a * f).|| = |.a.| * ||.f.||
proof
reconsider f1 = f, h1 = a * f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
A21: ( ( for s being Real st s in PreNorms h1 holds
s <= |.a.| * ||.f.|| ) implies upper_bound (PreNorms h1) <= |.a.| * ||.f.|| ) by SEQ_4:45;
A22: now :: thesis: for t being VECTOR of X
for s being VECTOR of Y st ||.t.|| <= 1 & ||.s.|| <= 1 holds
||.(h1 . (t,s)).|| <= |.a.| * ||.f.||
A23: 0 <= ||.f.|| by Th33;
let t be VECTOR of X; :: thesis: for s being VECTOR of Y st ||.t.|| <= 1 & ||.s.|| <= 1 holds
||.(h1 . (t,s)).|| <= |.a.| * ||.f.||
let s be VECTOR of Y; :: thesis: ( ||.t.|| <= 1 & ||.s.|| <= 1 implies ||.(h1 . (t,s)).|| <= |.a.| * ||.f.|| )
assume ( ||.t.|| <= 1 & ||.s.|| <= 1 ) ; :: thesis: ||.(h1 . (t,s)).|| <= |.a.| * ||.f.||
then ||.t.|| * ||.s.|| <= 1 * 1 by XREAL_1:66;
then A24: ||.f.|| * (||.t.|| * ||.s.||) <= ||.f.|| * 1 by A23, XREAL_1:64;
||.(f1 . (t,s)).|| <= (||.f.|| * ||.t.||) * ||.s.|| by Th32;
then A25: ||.(f1 . (t,s)).|| <= ||.f.|| by A24, XXREAL_0:2;
A26: ||.(a * (f1 . (t,s))).|| = |.a.| * ||.(f1 . (t,s)).|| by NORMSP_1:def 1;
A27: 0 <= |.a.| by COMPLEX1:46;
||.(h1 . (t,s)).|| = ||.(a * (f1 . (t,s))).|| by Th36;
hence ||.(h1 . (t,s)).|| <= |.a.| * ||.f.|| by A25, A26, A27, XREAL_1:64; :: thesis: verum
end;
A28: 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 ex s being VECTOR of Y st
( r = ||.(h1 . (t,s)).|| & ||.t.|| <= 1 & ||.s.|| <= 1 ) ;
hence r <= |.a.| * ||.f.|| by A22; :: thesis: verum
end;
A29: now :: thesis: ( ( a <> 0 & |.a.| * ||.f.|| <= ||.(a * f).|| ) or ( a = 0 & |.a.| * ||.f.|| = ||.(a * f).|| ) )
per cases ( a <> 0 or a = 0 ) ;
case A30: a <> 0 ; :: thesis: |.a.| * ||.f.|| <= ||.(a * f).||
A31: now :: thesis: for t being VECTOR of X
for s being VECTOR of Y st ||.t.|| <= 1 & ||.s.|| <= 1 holds
||.(f1 . (t,s)).|| <= (|.a.| ") * ||.(a * f).||
A32: 0 <= ||.(a * f).|| by Th33;
let t be VECTOR of X; :: thesis: for s being VECTOR of Y st ||.t.|| <= 1 & ||.s.|| <= 1 holds
||.(f1 . (t,s)).|| <= (|.a.| ") * ||.(a * f).||
let s be VECTOR of Y; :: thesis: ( ||.t.|| <= 1 & ||.s.|| <= 1 implies ||.(f1 . (t,s)).|| <= (|.a.| ") * ||.(a * f).|| )
assume ( ||.t.|| <= 1 & ||.s.|| <= 1 ) ; :: thesis: ||.(f1 . (t,s)).|| <= (|.a.| ") * ||.(a * f).||
then ||.t.|| * ||.s.|| <= 1 * 1 by XREAL_1:66;
then A33: ||.(a * f).|| * (||.t.|| * ||.s.||) <= ||.(a * f).|| * 1 by A32, XREAL_1:64;
||.(h1 . (t,s)).|| <= (||.(a * f).|| * ||.t.||) * ||.s.|| by Th32;
then A34: ||.(h1 . (t,s)).|| <= ||.(a * f).|| by A33, XXREAL_0:2;
h1 . (t,s) = a * (f1 . (t,s)) by Th36;
then A35: (a ") * (h1 . (t,s)) = ((a ") * a) * (f1 . (t,s)) by RLVECT_1:def 7
.= 1 * (f1 . (t,s)) by A30, XCMPLX_0:def 7
.= f1 . (t,s) by RLVECT_1:def 8 ;
A36: |.(a ").| = |.(1 * (a ")).|
.= |.(1 / a).| by XCMPLX_0:def 9
.= 1 / |.a.| by ABSVALUE:7
.= 1 * (|.a.| ") by XCMPLX_0:def 9
.= |.a.| " ;
A37: 0 <= |.(a ").| by COMPLEX1:46;
||.((a ") * (h1 . (t,s))).|| = |.(a ").| * ||.(h1 . (t,s)).|| by NORMSP_1:def 1;
hence ||.(f1 . (t,s)).|| <= (|.a.| ") * ||.(a * f).|| by A34, A35, A36, A37, XREAL_1:64; :: thesis: verum
end;
A38: 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 ex s being VECTOR of Y st
( r = ||.(f1 . (t,s)).|| & ||.t.|| <= 1 & ||.s.|| <= 1 ) ;
hence r <= (|.a.| ") * ||.(a * f).|| by A31; :: thesis: verum
end;
A39: ( ( 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;
A40: 0 <= |.a.| by COMPLEX1:46;
||.f.|| <= (|.a.| ") * ||.(a * f).|| by A38, A39, Th30;
then |.a.| * ||.f.|| <= |.a.| * ((|.a.| ") * ||.(a * f).||) by A40, XREAL_1:64;
then A41: |.a.| * ||.f.|| <= (|.a.| * (|.a.| ")) * ||.(a * f).|| ;
|.a.| <> 0 by A30, COMPLEX1:47;
then |.a.| * ||.f.|| <= 1 * ||.(a * f).|| by A41, XCMPLX_0:def 7;
hence |.a.| * ||.f.|| <= ||.(a * f).|| ; :: thesis: verum
end;
end;
end;
||.(a * f).|| <= |.a.| * ||.f.|| by A21, A28, Th30;
hence ||.(a * f).|| = |.a.| * ||.f.|| by A29, XXREAL_0:1; :: thesis: verum
end;
now :: thesis: ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) )
reconsider g = f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
set z = the carrier of [:X,Y:] --> (0. Z);
reconsider z = the carrier of [:X,Y:] --> (0. Z) as Function of the carrier of [:X,Y:], the carrier of Z ;
assume A44: ||.f.|| = 0 ; :: thesis: f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z))
now :: thesis: for t being VECTOR of X
for s being VECTOR of Y holds g . (t,s) = z . (t,s)
let t be VECTOR of X; :: thesis: for s being VECTOR of Y holds g . (t,s) = z . (t,s)
let s be VECTOR of Y; :: thesis: g . (t,s) = z . (t,s)
A45: [t,s] is Point of [:X,Y:] ;
||.(g . (t,s)).|| <= (||.f.|| * ||.t.||) * ||.s.|| by Th32;
then ||.(g . (t,s)).|| = 0 by A44;
hence g . (t,s) = 0. Z by NORMSP_0:def 5
.= z . (t,s) by A45, FUNCOP_1:7 ;
:: thesis: verum
end;
then g = z by BINOP_1:2;
hence f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) by Th31; :: thesis: verum
end;
hence ( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ) & ( f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) implies ||.f.|| = 0 ) & ||.(a * f).|| = |.a.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| ) by A1, A9, A20; :: thesis: verum