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 & Z is closed holds
I .: Z is closed
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 & Z is closed holds
I .: Z is closed
let Z be Subset of S; :: thesis: ( I is one-to-one & I is onto & I is isometric & Z is closed implies I .: Z is closed )
assume that
A1: ( I is one-to-one & I is onto ) and
A2: I is isometric and
A3: Z is closed ; :: thesis: I .: Z is closed
P1: dom I = the carrier of S by FUNCT_2:def 1;
consider J being LinearOperator of T,S such that
P2: ( J = I " & J is one-to-one & J is onto & J is isometric ) by A1, A2, LM020;
now :: thesis: for t1 being sequence of T st rng t1 c= I .: Z & t1 is convergent holds
lim t1 in I .: Z
let t1 be sequence of T; :: thesis: ( rng t1 c= I .: Z & t1 is convergent implies lim t1 in I .: Z )
assume A4: ( rng t1 c= I .: Z & t1 is convergent ) ; :: thesis: lim t1 in I .: Z
then A5: J * t1 is convergent by P2, LM022;
A6: rng (J * t1) = J .: (rng t1) by RELAT_1:127;
J .: (I .: Z) = I " (I .: Z) by A1, P2, FUNCT_1:85
.= Z by FUNCT_1:94, P1, A1 ;
then lim (J * t1) in Z by A3, A4, A5, A6, NFCONT_1:def 3, RELAT_1:123;
then J . (lim t1) in Z by A4, P2, LM021;
then I . (J . (lim t1)) in I .: Z by FUNCT_2:35;
hence lim t1 in I .: Z by A1, P2, FUNCT_1:35; :: thesis: verum
end;
hence I .: Z is closed by NFCONT_1:def 3; :: thesis: verum