let X, Y be ComplexNormSpace; :: thesis: for g being LinearOperator of X,Y holds

( g is Lipschitzian iff PreNorms g is bounded_above )

let g be LinearOperator of X,Y; :: thesis: ( g is Lipschitzian iff PreNorms g is bounded_above )

( g is Lipschitzian iff PreNorms g is bounded_above )

let g be LinearOperator of X,Y; :: thesis: ( g is Lipschitzian iff PreNorms g is bounded_above )

now :: thesis: ( PreNorms g is bounded_above implies ex K being Real st g is Lipschitzian )

hence
( g is Lipschitzian iff PreNorms g is bounded_above )
by Th26; :: thesis: verumreconsider K = upper_bound (PreNorms g) as Real ;

assume A1: PreNorms g is bounded_above ; :: thesis: ex K being Real st g is Lipschitzian

0 <= K

end;assume A1: PreNorms g is bounded_above ; :: thesis: ex K being Real st g is Lipschitzian

A2: now :: thesis: for t being VECTOR of X holds ||.(g . t).|| <= K * ||.t.||

take K = K; :: thesis: g is Lipschitzian let t be VECTOR of X; :: thesis: ||.(g . t).|| <= K * ||.t.||

end;now :: thesis: ( ( t = 0. X & ||.(g . t).|| <= K * ||.t.|| ) or ( t <> 0. X & ||.(g . t).|| <= K * ||.t.|| ) )end;

hence
||.(g . t).|| <= K * ||.t.||
; :: thesis: verumper cases
( t = 0. X or t <> 0. X )
;

end;

case A3:
t = 0. X
; :: thesis: ||.(g . t).|| <= K * ||.t.||

then A4:
||.t.|| = 0
by NORMSP_0:def 6;

g . t = g . (0c * (0. X)) by A3, CLVECT_1:1

.= 0c * (g . (0. X)) by Def3

.= 0. Y by CLVECT_1:1 ;

hence ||.(g . t).|| <= K * ||.t.|| by A4, NORMSP_0:def 6; :: thesis: verum

end;g . t = g . (0c * (0. X)) by A3, CLVECT_1:1

.= 0c * (g . (0. X)) by Def3

.= 0. Y by CLVECT_1:1 ;

hence ||.(g . t).|| <= K * ||.t.|| by A4, NORMSP_0:def 6; :: thesis: verum

case A5:
t <> 0. X
; :: thesis: ||.(g . t).|| <= K * ||.t.||

reconsider t0 = (||.t.|| ") + (0 * <i>) as Element of COMPLEX by XCMPLX_0:def 2;

reconsider t1 = t0 * t as VECTOR of X ;

A6: ||.t.|| <> 0 by A5, NORMSP_0:def 5;

then A7: ||.t.|| > 0 by CLVECT_1:105;

A8: |.((||.t.|| ") + (0 * <i>)).| = |.(1 * (||.t.|| ")).|

.= |.(1 / ||.t.||).| by XCMPLX_0:def 9

.= 1 / |.||.t.||.| by ABSVALUE:7

.= 1 / ||.t.|| by A7, ABSVALUE:def 1

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

.= ||.t.|| " ;

then A9: ||.(g . t).|| / ||.t.|| = ||.(g . t).|| * |.t0.| by XCMPLX_0:def 9

.= ||.(t0 * (g . t)).|| by CLVECT_1:def 13

.= ||.(g . t1).|| by Def3 ;

||.t1.|| = |.t0.| * ||.t.|| by CLVECT_1:def 13

.= 1 by A6, A8, XCMPLX_0:def 7 ;

then ||.(g . t).|| / ||.t.|| in PreNorms g by A9;

then A10: ||.(g . t).|| / ||.t.|| <= K by A1, SEQ_4:def 1;

(||.(g . t).|| / ||.t.||) * ||.t.|| = (||.(g . t).|| * (||.t.|| ")) * ||.t.|| by XCMPLX_0:def 9

.= ||.(g . t).|| * ((||.t.|| ") * ||.t.||)

.= ||.(g . t).|| * 1 by A6, XCMPLX_0:def 7

.= ||.(g . t).|| ;

hence ||.(g . t).|| <= K * ||.t.|| by A7, A10, XREAL_1:64; :: thesis: verum

end;reconsider t1 = t0 * t as VECTOR of X ;

A6: ||.t.|| <> 0 by A5, NORMSP_0:def 5;

then A7: ||.t.|| > 0 by CLVECT_1:105;

A8: |.((||.t.|| ") + (0 * <i>)).| = |.(1 * (||.t.|| ")).|

.= |.(1 / ||.t.||).| by XCMPLX_0:def 9

.= 1 / |.||.t.||.| by ABSVALUE:7

.= 1 / ||.t.|| by A7, ABSVALUE:def 1

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

.= ||.t.|| " ;

then A9: ||.(g . t).|| / ||.t.|| = ||.(g . t).|| * |.t0.| by XCMPLX_0:def 9

.= ||.(t0 * (g . t)).|| by CLVECT_1:def 13

.= ||.(g . t1).|| by Def3 ;

||.t1.|| = |.t0.| * ||.t.|| by CLVECT_1:def 13

.= 1 by A6, A8, XCMPLX_0:def 7 ;

then ||.(g . t).|| / ||.t.|| in PreNorms g by A9;

then A10: ||.(g . t).|| / ||.t.|| <= K by A1, SEQ_4:def 1;

(||.(g . t).|| / ||.t.||) * ||.t.|| = (||.(g . t).|| * (||.t.|| ")) * ||.t.|| by XCMPLX_0:def 9

.= ||.(g . t).|| * ((||.t.|| ") * ||.t.||)

.= ||.(g . t).|| * 1 by A6, XCMPLX_0:def 7

.= ||.(g . t).|| ;

hence ||.(g . t).|| <= K * ||.t.|| by A7, A10, XREAL_1:64; :: thesis: verum

0 <= K

proof

hence
g is Lipschitzian
by A2; :: thesis: verum
consider r0 being object such that

A11: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A11;

hence 0 <= K by A1, A11, SEQ_4:def 1; :: thesis: verum

end;A11: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A11;

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

0 <= r

then
0 <= r0
by A11;0 <= r

let r be Real; :: thesis: ( r in PreNorms g implies 0 <= r )

assume r in PreNorms g ; :: thesis: 0 <= r

then ex t being VECTOR of X st

( r = ||.(g . t).|| & ||.t.|| <= 1 ) ;

hence 0 <= r by CLVECT_1:105; :: thesis: verum

end;assume r in PreNorms g ; :: thesis: 0 <= r

then ex t being VECTOR of X st

( r = ||.(g . t).|| & ||.t.|| <= 1 ) ;

hence 0 <= r by CLVECT_1:105; :: thesis: verum

hence 0 <= K by A1, A11, SEQ_4:def 1; :: thesis: verum