let X be RealNormSpace-Sequence; :: thesis: for Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedMultilinearOperators (X,Y)) holds 0 <= ||.f.||
let Y be RealNormSpace; :: thesis: for f being Point of (R_NormSpace_of_BoundedMultilinearOperators (X,Y)) holds 0 <= ||.f.||
let f be Point of (R_NormSpace_of_BoundedMultilinearOperators (X,Y)); :: thesis: 0 <= ||.f.||
reconsider g = f as Lipschitzian MultilinearOperator of X,Y by Def9;
consider r0 being object such that
A1: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A1;
A2: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th27;
A3: (BoundedMultilinearOperatorsNorm (X,Y)) . f = upper_bound (PreNorms g) by Th30;
now :: thesis: for r being Real st r in PreNorms g holds
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 (product X) st
( r = ||.(g . t).|| & ( for i being Element of dom X holds ||.(t . i).|| <= 1 ) ) ;
hence 0 <= r ; :: thesis: verum
end;
then 0 <= r0 by A1;
hence 0 <= ||.f.|| by A1, A2, A3, SEQ_4:def 1; :: thesis: verum