 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 Th20:
  for T be NormedLinearTopSpace holds
    T is compact iff TopSpaceNorm T is compact
proof
  let T be NormedLinearTopSpace;
  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;
A3: [#] T = [#](TopSpaceMetr MetricSpaceNorm T);
  hereby assume T is compact; then
    [#] T is compact Subset of TopSpaceMetr MetricSpaceNorm T
      by A2,COMPTS_1:23,COMPTS_1:1;
    hence TopSpaceNorm T is compact by A3,COMPTS_1:1;
  end;
  assume TopSpaceNorm T is compact; then
  [#] T is compact Subset of TopSpaceMetr MetricSpaceNorm T by A3,COMPTS_1:1;
  hence T is compact by A2,COMPTS_1:1,COMPTS_1:23;
end;
