Approximately additive Schwartz distributions

Generalizing the Hyers–Ulam–Rassias stability theorem [Th.M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300] to the space of Schwartz distributions, we introduce a concept of approximately additive Schwartz distributions and prove that every approximately additive distribution can be approximated by linear functions.