 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem Th27:
for X being RealUnitarySpace,
    M be Subspace of X,
    K be Subset of X,
    L be Subset of TopSpaceNorm RUSp2RNSp X
st the carrier of M = L & K = Cl(L) holds
      K is linearly-closed
proof
let X being RealUnitarySpace,
    M be Subspace of X,
    K be Subset of X,
    L be Subset of TopSpaceNorm RUSp2RNSp X;
assume A1: the carrier of M = L & K = Cl(L);
reconsider CM = the carrier of M as
  Subset of X by RUSUB_1:def 1;
A3: L c= Cl(L) by PRE_TOPC:18;
A4:for v, u being VECTOR of X st
   v in K & u in K holds v + u in K
proof
  let v, u be VECTOR of X;
  assume A5p:v in K & u in K; then
consider S being sequence of X such that
 A6: ( for n being Nat holds S . n in L )
    & S is convergent
    & lim S = v by Th10,A1;
consider T being sequence of X such that
 A7: ( for n being Nat holds T . n in L )
    & T is convergent
    & lim T = u by Th10,A5p,A1;
A8: for n be Nat holds (S+T).n in Cl(L)
proof
  let n be Nat;
  A9: (S+T).n = S.n + T.n by NORMSP_1:def 2;
   S.n in M & T.n in M by A1,STRUCT_0:def 5,A6,A7;
   then
   S.n + T.n in L by A1,STRUCT_0:def 5,RUSUB_1:14;
   hence (S+T).n in Cl(L) by A9,A3;
end;
lim (S+T) = v+u by BHSP_2:13,A6,A7;
hence thesis by A8,A1,BHSP_2:3,A6,A7,Th11;
end;
for a being Real
         for v being VECTOR of X st v in K
       holds a * v in K
proof
let a be Real;
let v be VECTOR of X;
 assume v in K; then
consider S being sequence of X such that
 A11: ( for n being Nat holds S . n in L)
    & S is convergent
    & lim S = v by Th10,A1;
A12: for n be Nat holds (a*S).n in Cl(L)
proof
  let n be Nat;
  A13: (a*S).n = a*S.n by NORMSP_1:def 5;
   S.n in M by A1,STRUCT_0:def 5,A11; then
   a*S.n in L by A1,STRUCT_0:def 5,RUSUB_1:15;
   hence (a*S).n in Cl(L) by A13,A3;
end;
lim (a*S) = a*v by BHSP_2:15,A11;
hence thesis by A12,A1,BHSP_2:5,A11,Th11;
end;
hence thesis by A4,RLSUB_1:def 1;
end;
