
theorem Th24:
  for X,Y be RealNormSpace holds
  BoundedMultilinearOperatorsNorm(<*X*>,Y)
    = BoundedLinearOperatorsNorm(product <*X*>,Y)
  proof
    let X,Y be RealNormSpace;
    set n1 = BoundedMultilinearOperatorsNorm(<*X*>,Y);
    set n2 = BoundedLinearOperatorsNorm(product <*X*>,Y);
    A1: BoundedMultilinearOperators(<*X*>,Y)
      = BoundedLinearOperators(product <*X*>,Y) by Th23;
    A2: dom <*X*> = {1} by FINSEQ_1:2,38; then
    reconsider i = 1 as Element of dom <*X*> by TARSKI:def 1;

    for f be object st f in BoundedMultilinearOperators(<*X*>,Y)
    holds n1.f = n2.f
    proof
      let f be object;
      assume
      A4: f in BoundedMultilinearOperators(<*X*>,Y);
      reconsider f1 = f
        as Lipschitzian MultilinearOperator of <*X*>,Y
        by A4,LOPBAN10:def 11;
      A5: modetrans(f,<*X*>,Y) = f1;
      A6: n1.f = upper_bound PreNorms(f1) by A4,A5,LOPBAN10:def 15;
      reconsider f2 = f
        as Lipschitzian LinearOperator of product <*X*>,Y
        by A1,A4,LOPBAN_1:def 9;
      A7: modetrans(f,product <*X*>,Y) = f2 by A1,A4,LOPBAN_1:def 11;
      A8: n2.f = upper_bound(PreNorms(f2)) by A1,A4,A7,LOPBAN_1:def 13;

      for n be object holds n in PreNorms(f1) iff n in PreNorms(f2)
      proof
        let n be object;
        hereby
          assume n in PreNorms(f1); then
          consider x be VECTOR of product <*X*> such that
          A9: n = ||.f1.x.||
            & for i be Element of dom <*X*> holds ||.x.i.|| <= 1;
          consider x1 be Point of X such that
          A10: x = <*x1*> by Th12;
          A11: ||.x.|| = ||.x1.|| by A10,Th12;
          ||.x.i.|| <= 1 by A9; then
          ||.x1.|| <= 1 by A10;
          hence n in PreNorms(f2) by A9,A11;
        end;

        assume n in PreNorms(f2); then
        consider s be VECTOR of product <*X*> such that
        A12: n = ||.f2.s.|| & ||.s.|| <= 1;
        consider s1 be Point of X such that
        A13: s = <*s1*> by Th12;
        A14: ||.s.|| = ||.s1.|| by A13,Th12;

        for i be Element of dom <*X*> holds ||.s.i.|| <= 1
        proof
          let i be Element of dom <*X*>;
          A15: i = 1 by A2,TARSKI:def 1;
          thus ||.s.i.|| <= 1 by A12,A13,A14,A15;
        end;
        hence n in PreNorms(f1) by A12;
      end;
      hence n1.f = n2.f by A6,A8,TARSKI:2;
    end;
    hence n1 = n2 by Th23;
  end;
