let S, T be RealNormSpace; :: thesis: for I being LinearOperator of S,T
for s1 being sequence of S st I is one-to-one & I is onto & I is isometric holds
( s1 is convergent iff I * s1 is convergent )
let I be LinearOperator of S,T; :: thesis: for s1 being sequence of S st I is one-to-one & I is onto & I is isometric holds
( s1 is convergent iff I * s1 is convergent )
let s1 be sequence of S; :: thesis: ( I is one-to-one & I is onto & I is isometric implies ( s1 is convergent iff I * s1 is convergent ) )
assume AS: ( I is one-to-one & I is onto & I is isometric ) ; :: thesis: ( s1 is convergent iff I * s1 is convergent )
consider J being LinearOperator of T,S such that
P2: ( J = I " & J is one-to-one & J is onto & J is isometric ) by AS, LM020;
thus ( s1 is convergent implies I * s1 is convergent ) by LM021, AS; :: thesis: ( I * s1 is convergent implies s1 is convergent )
assume P4: I * s1 is convergent ; :: thesis: s1 is convergent
P6: rng s1 c= the carrier of S ;
J * (I * s1) = (J * I) * s1 by RELAT_1:36
.= (id the carrier of S) * s1 by AS, P2, FUNCT_2:29
.= s1 by RELAT_1:53, P6 ;
hence s1 is convergent by P2, P4, LM021; :: thesis: verum