let S, T be RealNormSpace; :: thesis: for I being LinearOperator of S,T
for Z being Subset of S st I is one-to-one & I is onto & I is isometric-like holds
( Z is compact iff I .: Z is compact )
let I be LinearOperator of S,T; :: thesis: for Z being Subset of S st I is one-to-one & I is onto & I is isometric-like holds
( Z is compact iff I .: Z is compact )
let Z be Subset of S; :: thesis: ( I is one-to-one & I is onto & I is isometric-like implies ( Z is compact iff I .: Z is compact ) )
assume that
A1: ( I is one-to-one & I is onto ) and
A2: I is isometric-like ; :: thesis: ( Z is compact iff I .: Z is compact )
consider J being LinearOperator of T,S such that
A3: ( J = I " & J is one-to-one & J is onto & J is isometric-like ) by A1, A2, Th29;
A4: dom I = the carrier of S by FUNCT_2:def 1;
thus ( Z is compact implies I .: Z is compact ) by A1, Lm5, A2; :: thesis: ( I .: Z is compact implies Z is compact )
thus ( I .: Z is compact implies Z is compact ) :: thesis: verum

proof

assume A5: I .: Z is compact ; :: thesis: Z is compact
J .: (I .: Z) = I " (I .: Z) by A1, A3, FUNCT_1:85
.= Z by A1, A4, FUNCT_1:94 ;
hence Z is compact by A3, A5, Lm5; :: thesis: verum

end;