Characterization for Beurling-Bjorck space and Schwartz space

We give an elementary proof of the equivalence of the original definition of Schwartz and our characterization for the Schwartz space S. The new proof is based on the Landau inequality concerning the estimates of derivatives. Applying the same method, as an application, we give a better symmetric characterization of the Beurling-Bjorck space of test functions for tempered ultradistributions with respect to Fourier transform without conditions on derivatives.