A short proof of the Arens-Eells embedding theorem
E. Michael · Proceedings of the American Mathematical Society · 1964
THE ARENS-EELLS EMBEDDING THEOREM 415 5. Proof of Example 1.4.If m were continuous, there would exist a compact K EX such that / sufficiently small on K implies (5.1) | [«(/)](y) | <1 for all yE Y.However, for any such K there exists a yoG Y-p(K),and if we pick fEC(X, R) to be 0 on K and 2 on p-1(yo),then [u(f)](y0) = 2, and hence/ fails to satisfy (5.1).This implies that m cannot be continuous.