New definition of products of distributions (Q2724779)

The author suggests to define the product of two distributions \(S,T\in{\mathcal D}'(\mathbb{R}^n)\) as a nonlinear functional on \({\mathcal D}(\mathbb{R}^n)\) by letting \((ST)(\varphi):= S(\varphi)T(\varphi)\), \(\varphi\in{\mathcal D}(\mathbb{R}^n)\). His suggestion means to embedd \({\mathcal D}'(\mathbb{R}^n)\) into the algebra \(C({\mathcal D}(\mathbb{R}^n))\) of all continuous functions on \({\mathcal D}(\mathbb{R}^n)\) and to use the multiplication defined there.