let X be non empty set ; :: thesis: for Y being ComplexNormSpace

for f, g being Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y)); :: 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.|| )

for f, g being Point of (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y))

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 (C_NormSpace_of_BoundedFunctions (X,Y)); :: 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 :: thesis: ( f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) implies ||.f.|| = 0 )

A9:
||.(f + g).|| <= ||.f.|| + ||.g.||
assume A2:
f = 0. (C_NormSpace_of_BoundedFunctions (X,Y))
; :: thesis: ||.f.|| = 0

thus ||.f.|| = 0 :: thesis: verum

end;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 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;

A5: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th12;

A6: z = g by A2, Th16;

then upper_bound (PreNorms g) = 0 by A7, A5, A3, A4, SEQ_4:def 1;

then (ComplexBoundedFunctionsNorm (X,Y)) . f = 0 by Th15;

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

end;set z = X --> (0. Y);

reconsider z = X --> (0. Y) as Function of X, the carrier of Y ;

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;

A5: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th12;

A6: z = g by A2, Th16;

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 Element of X such that

A8: r = ||.(g . t).|| ;

||.(g . t).|| = ||.(0. Y).|| by A6, FUNCOP_1:7

.= 0 ;

hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum

end;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_1:7

.= 0 ;

hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum

then upper_bound (PreNorms g) = 0 by A7, A5, A3, A4, SEQ_4:def 1;

then (ComplexBoundedFunctionsNorm (X,Y)) . f = 0 by Th15;

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

proof

A14:
||.(c * f).|| = |.c.| * ||.f.||
reconsider f1 = f, g1 = g, h1 = f + g as bounded Function of X, the carrier of Y by Def5;

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;

hence ||.(f + g).|| <= ||.f.|| + ||.g.|| by A13, A10; :: thesis: verum

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 Element of X holds ||.(h1 . t).|| <= ||.f.|| + ||.g.||

let t be Element of X; :: thesis: ||.(h1 . t).|| <= ||.f.|| + ||.g.||

( ||.(f1 . t).|| <= ||.f.|| & ||.(g1 . t).|| <= ||.g.|| ) by Th17;

then A12: ||.(f1 . t).|| + ||.(g1 . t).|| <= ||.f.|| + ||.g.|| by XREAL_1:7;

( ||.(h1 . t).|| = ||.((f1 . t) + (g1 . t)).|| & ||.((f1 . t) + (g1 . t)).|| <= ||.(f1 . t).|| + ||.(g1 . t).|| ) by Th20, CLVECT_1:def 13;

hence ||.(h1 . t).|| <= ||.f.|| + ||.g.|| by A12, XXREAL_0:2; :: thesis: verum

end;( ||.(f1 . t).|| <= ||.f.|| & ||.(g1 . t).|| <= ||.g.|| ) by Th17;

then A12: ||.(f1 . t).|| + ||.(g1 . t).|| <= ||.f.|| + ||.g.|| by XREAL_1:7;

( ||.(h1 . t).|| = ||.((f1 . t) + (g1 . t)).|| & ||.((f1 . t) + (g1 . t)).|| <= ||.(f1 . t).|| + ||.(g1 . t).|| ) by Th20, CLVECT_1:def 13;

hence ||.(h1 . t).|| <= ||.f.|| + ||.g.|| by A12, XXREAL_0:2; :: thesis: verum

A13: now :: thesis: for r being Real st r in PreNorms h1 holds

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

(ComplexBoundedFunctionsNorm (X,Y)) . (f + g) = upper_bound (PreNorms h1)
by Th15;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 Element of X st r = ||.(h1 . t).|| ;

hence r <= ||.f.|| + ||.g.|| by A11; :: thesis: verum

end;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

hence ||.(f + g).|| <= ||.f.|| + ||.g.|| by A13, A10; :: thesis: verum

proof

reconsider f1 = f, h1 = c * f as bounded Function of X, the carrier of Y by Def5;

A15: ( ( for s being Real st s in PreNorms h1 holds

s <= |.c.| * ||.f.|| ) implies upper_bound (PreNorms h1) <= |.c.| * ||.f.|| ) by SEQ_4:45;

then ||.(c * f).|| <= |.c.| * ||.f.|| by A18, A15;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A19, XXREAL_0:1; :: thesis: verum

end;A15: ( ( for s being Real st s in PreNorms h1 holds

s <= |.c.| * ||.f.|| ) implies upper_bound (PreNorms h1) <= |.c.| * ||.f.|| ) by SEQ_4:45;

A16: now :: thesis: for t being Element of X holds ||.(h1 . t).|| <= |.c.| * ||.f.||

let t be Element of X; :: thesis: ||.(h1 . t).|| <= |.c.| * ||.f.||

A17: 0 <= |.c.| by COMPLEX1:46;

( ||.(h1 . t).|| = ||.(c * (f1 . t)).|| & ||.(c * (f1 . t)).|| = |.c.| * ||.(f1 . t).|| ) by Th21, CLVECT_1:def 13;

hence ||.(h1 . t).|| <= |.c.| * ||.f.|| by A17, Th17, XREAL_1:64; :: thesis: verum

end;A17: 0 <= |.c.| by COMPLEX1:46;

( ||.(h1 . t).|| = ||.(c * (f1 . t)).|| & ||.(c * (f1 . t)).|| = |.c.| * ||.(f1 . t).|| ) by Th21, CLVECT_1:def 13;

hence ||.(h1 . t).|| <= |.c.| * ||.f.|| by A17, Th17, XREAL_1:64; :: thesis: verum

A18: now :: thesis: for r being Real st r in PreNorms h1 holds

r <= |.c.| * ||.f.||

r <= |.c.| * ||.f.||

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;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

A19: now :: thesis: ( ( c <> 0c & |.c.| * ||.f.|| <= ||.(c * f).|| ) or ( c = 0c & ||.(c * f).|| = |.c.| * ||.f.|| ) )end;

(ComplexBoundedFunctionsNorm (X,Y)) . (c * f) = upper_bound (PreNorms h1)
by Th15;per cases
( c <> 0c or c = 0c )
;

end;

case A20:
c <> 0c
; :: thesis: |.c.| * ||.f.|| <= ||.(c * f).||

s <= (|.c.| ") * ||.(c * f).|| ) implies upper_bound (PreNorms f1) <= (|.c.| ") * ||.(c * f).|| ) by SEQ_4:45;

A26: 0 <= |.c.| by COMPLEX1:46;

(ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f1) by Th15;

then ||.f.|| <= (|.c.| ") * ||.(c * f).|| by A24, A25;

then |.c.| * ||.f.|| <= |.c.| * ((|.c.| ") * ||.(c * f).||) by A26, XREAL_1:64;

then A27: |.c.| * ||.f.|| <= (|.c.| * (|.c.| ")) * ||.(c * f).|| ;

|.c.| <> 0 by A20, COMPLEX1:47;

then |.c.| * ||.f.|| <= 1 * ||.(c * f).|| by A27, XCMPLX_0:def 7;

hence |.c.| * ||.f.|| <= ||.(c * f).|| ; :: thesis: verum

end;

A21: now :: thesis: for t being Element of X holds ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).||

let t be Element of X; :: thesis: ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).||

A22: |.(c ").| = |.(1r / c).| by COMPLEX1:def 4, XCMPLX_1:215

.= 1 / |.c.| by COMPLEX1:48, COMPLEX1:67

.= 1 * (|.c.| ") by XCMPLX_0:def 9

.= |.c.| " ;

h1 . t = c * (f1 . t) by Th21;

then A23: (c ") * (h1 . t) = ((c ") * c) * (f1 . t) by CLVECT_1:def 4

.= 1r * (f1 . t) by A20, COMPLEX1:def 4, XCMPLX_0:def 7

.= f1 . t by CLVECT_1:def 5 ;

( ||.((c ") * (h1 . t)).|| = |.(c ").| * ||.(h1 . t).|| & 0 <= |.(c ").| ) by CLVECT_1:def 13, COMPLEX1:46;

hence ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).|| by A23, A22, Th17, XREAL_1:64; :: thesis: verum

end;A22: |.(c ").| = |.(1r / c).| by COMPLEX1:def 4, XCMPLX_1:215

.= 1 / |.c.| by COMPLEX1:48, COMPLEX1:67

.= 1 * (|.c.| ") by XCMPLX_0:def 9

.= |.c.| " ;

h1 . t = c * (f1 . t) by Th21;

then A23: (c ") * (h1 . t) = ((c ") * c) * (f1 . t) by CLVECT_1:def 4

.= 1r * (f1 . t) by A20, COMPLEX1:def 4, XCMPLX_0:def 7

.= f1 . t by CLVECT_1:def 5 ;

( ||.((c ") * (h1 . t)).|| = |.(c ").| * ||.(h1 . t).|| & 0 <= |.(c ").| ) by CLVECT_1:def 13, COMPLEX1:46;

hence ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).|| by A23, A22, Th17, XREAL_1:64; :: thesis: verum

A24: now :: thesis: for r being Real st r in PreNorms f1 holds

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

A25:
( ( for s being Real st s in PreNorms f1 holds r <= (|.c.| ") * ||.(c * f).||

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;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

s <= (|.c.| ") * ||.(c * f).|| ) implies upper_bound (PreNorms f1) <= (|.c.| ") * ||.(c * f).|| ) by SEQ_4:45;

A26: 0 <= |.c.| by COMPLEX1:46;

(ComplexBoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f1) by Th15;

then ||.f.|| <= (|.c.| ") * ||.(c * f).|| by A24, A25;

then |.c.| * ||.f.|| <= |.c.| * ((|.c.| ") * ||.(c * f).||) by A26, XREAL_1:64;

then A27: |.c.| * ||.f.|| <= (|.c.| * (|.c.| ")) * ||.(c * f).|| ;

|.c.| <> 0 by A20, COMPLEX1:47;

then |.c.| * ||.f.|| <= 1 * ||.(c * f).|| by A27, XCMPLX_0:def 7;

hence |.c.| * ||.f.|| <= ||.(c * f).|| ; :: thesis: verum

case A28:
c = 0c
; :: thesis: ||.(c * f).|| = |.c.| * ||.f.||

reconsider fz = f as VECTOR of (C_VectorSpace_of_BoundedFunctions (X,Y)) ;

c * f = (Mult_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))) . [c,f] by CLVECT_1:def 1

.= c * fz by CLVECT_1:def 1

.= 0. (C_VectorSpace_of_BoundedFunctions (X,Y)) by A28, CLVECT_1:1

.= 0. (C_NormSpace_of_BoundedFunctions (X,Y)) ;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A28, Th19, COMPLEX1:44; :: thesis: verum

end;c * f = (Mult_ ((ComplexBoundedFunctions (X,Y)),(ComplexVectSpace (X,Y)))) . [c,f] by CLVECT_1:def 1

.= c * fz by CLVECT_1:def 1

.= 0. (C_VectorSpace_of_BoundedFunctions (X,Y)) by A28, CLVECT_1:1

.= 0. (C_NormSpace_of_BoundedFunctions (X,Y)) ;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A28, Th19, COMPLEX1:44; :: thesis: verum

then ||.(c * f).|| <= |.c.| * ||.f.|| by A18, A15;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A19, XXREAL_0:1; :: thesis: verum

now :: thesis: ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) )

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: verumreconsider 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))

hence f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) by Th16; :: thesis: verum

end;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 :: thesis: for t being Element of X holds g . t = z . t

then
g = z
by FUNCT_2:63;let t be Element of X; :: thesis: g . t = z . t

||.(g . t).|| <= ||.f.|| by Th17;

then ||.(g . t).|| = 0 by A29, CLVECT_1:105;

hence g . t = 0. Y by NORMSP_0:def 5

.= z . t by FUNCOP_1:7 ;

:: thesis: verum

end;||.(g . t).|| <= ||.f.|| by Th17;

then ||.(g . t).|| = 0 by A29, CLVECT_1:105;

hence g . t = 0. Y by NORMSP_0:def 5

.= z . t by FUNCOP_1:7 ;

:: thesis: verum

hence f = 0. (C_NormSpace_of_BoundedFunctions (X,Y)) by Th16; :: thesis: verum