
theorem Th75A:
for X be RealNormSpace holds
  the carrier of DualSp X
    = the carrier of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
proof
   let X be RealNormSpace;
A1:for x be object holds
    x in BoundedLinearFunctionals X iff x in BoundedLinearOperators(X,RNS_Real)
   proof
    let x be object;
    hereby assume x in BoundedLinearFunctionals X; then
     x is Lipschitzian additive homogeneous Functional of X
         by DUALSP01:def 10; then
     x is Lipschitzian additive homogeneous Function of X,RNS_Real by LMN7;
     hence x in BoundedLinearOperators(X,RNS_Real) by LOPBAN_1:def 9;
    end;
    assume x in BoundedLinearOperators(X,RNS_Real); then
    x is Lipschitzian LinearOperator of X,RNS_Real by LOPBAN_1:def 9; then
    x is Lipschitzian additive homogeneous Functional of X by LMN7;
    hence x in BoundedLinearFunctionals X by DUALSP01:def 10;
   end;
   R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
    = NORMSTR (# BoundedLinearOperators(X,RNS_Real),
          Zero_(BoundedLinearOperators(X,RNS_Real),
             R_VectorSpace_of_LinearOperators(X,RNS_Real)),
          Add_(BoundedLinearOperators(X,RNS_Real),
             R_VectorSpace_of_LinearOperators(X,RNS_Real)),
          Mult_(BoundedLinearOperators(X,RNS_Real),
             R_VectorSpace_of_LinearOperators(X,RNS_Real)),
          BoundedLinearOperatorsNorm(X,RNS_Real) #) by LOPBAN_1:def 14;
   hence thesis by A1,TARSKI:2;
end;
