ABSTRACT
In earlier chapters several spaces of distributions were defined; and the space D ′ turned out to be the largest of all spaces considered so far, except for the space Z′. It is possible to introduce spaces of generalized functions which are larger than D ′ . In order to retain the interpretation of generalized function spaces as duals of test function spaces, we have to construct test function spaces which are contained in D . It is interesting to note that if the test function spaces are some classes of non-quasi-analytic functions with some natural topology, then the dual spaces, have nice properties, analogous to those of distributions. The elements of these dual spaces are called ultradistributions, see for instance Lions and Magenes (1973). We should not confuse this with our earlier terminology used in Section 1.9.1 where the elements of Z′ are called ultradistributions, which has also been used by Gel’fand and Shilov (1964) and Zemanian (1965).
