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Mar 30 at 18:37 comment added TaQ It is also in Horváth's book Topological Vector Spaces and Distribution as Proposition 4.11.5 on page 417 with a 1½ page long proof.
Mar 30 at 17:02 comment added Liding Yao @JochenWengenroth Thanks! I didn't locate the index of theorem for which characterize $\mathscr O_M$ in his book (since I don't know French:( ). Could you help pointing that?
Mar 30 at 6:50 comment added Jochen Wengenroth This is true (up to a multiplicative constant). It is in Laurent Schwartz's famous book Théorie des distributions. He called this multiplier space $\mathscr O_M(\mathbb R^n)$.
Mar 30 at 3:08 history asked Liding Yao CC BY-SA 4.0