Characterization for Beurling-Björck 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 . 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-Björck space of test functions for tempered ultradistributions with respect to Fourier transform without conditions on derivatives.