let X, Y be RealNormSpace; :: thesis: for g being bounded LinearOperator of X,Y holds
( not PreNorms g is empty & PreNorms g is bounded_above )
let g be bounded LinearOperator of X,Y; :: thesis: ( not PreNorms g is empty & PreNorms g is bounded_above )
PreNorms g is bounded_above

proof

consider K being Real such that
A1: 0 <= K and
A2: for x being VECTOR of X holds ||.(g . x).|| <= K * ||.x.|| by Def9;
take K ; :: according to SEQ_4:def 1 :: thesis: for b₁ being set holds
( not b₁ in PreNorms g or b₁ <= K )

now

let r be Real; :: thesis: ( r in PreNorms g implies r <= K )
assume r in PreNorms g ; :: thesis: r <= K
then consider t being VECTOR of X such that
A3: r = ||.(g . t).|| and
A4: ||.t.|| <= 1 ;
A5: ||.(g . t).|| <= K * ||.t.|| by A2;
K * ||.t.|| <= K * 1 by A1, A4, XREAL_1:66;
hence r <= K by A3, A5, XXREAL_0:2; :: thesis: verum

end;

hence for b₁ being set holds
( not b₁ in PreNorms g or b₁ <= K ) ; :: thesis: verum

end;

hence ( not PreNorms g is empty & PreNorms g is bounded_above ) ; :: thesis: verum