 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);
 reserve S,Z for RealNormSpace;
 reserve T for RealBanachSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th21:
  for T be NormedLinearTopSpace,
      X be set holds
  X is compact Subset of T
    iff
  X is compact Subset of TopSpaceNorm T
proof
  let T be NormedLinearTopSpace, X be set;
  consider RNS be RealNormSpace such that
A1: RNS = the NORMSTR of T
     & the topology of T = the topology of TopSpaceNorm RNS by C0SP3:def 6;
A2: TopSpaceMetr MetricSpaceNorm T
    = TopStruct(# the carrier of T, the topology of T #) by A1,Th15;
  hence X is compact Subset of T implies
  X is compact Subset of TopSpaceNorm T by COMPTS_1:23;
  assume X is compact Subset of TopSpaceNorm T; then
  reconsider X0=X as compact Subset of TopSpaceNorm T;
  X0 is compact Subset of TopStruct(# the carrier of T, the topology of T #)
    by A2;
  hence X is compact Subset of T by COMPTS_1:23;
end;
