Exercise 6, § 3

Let A be a unital commutative algebra over $\mathbf{C}$ such that the following conditions are satisfied:

(i) The algebra $\mathscr{C}(\mathbf{R})$ is a unital subalgebra of A;

(ii) There exists a derivation $d: A\rightarrow A$ whose restriction to $\mathscr{C}^1(\mathbf{R})$ coincides with the usual derivation of functions.

a) The identity mapping X of $\mathbf{R}$ is invertible in A. (Exercise prove that the mapping $g:\mathbf{R}\rightarrow \mathbf{R}$ such that $g(0) = 0$ and $g(x) =x$(log($|x|$)$-1$) for $x\not = 0$ is continuous and satisfies $gX = 1.$)

b) One has $d\circ d(|X|) = 0$.

c) There exists no bilinear mapping $m:\mathscr{D}'(\mathbf{R})\times \mathscr{D}'(\mathbf{R})\rightarrow \mathscr{D}'(\mathbf{R})$ such that $m(f, g) =f g$ for every pair $(f, g)\in \mathscr{D}(\mathbf{R})^2$, and such that

for every $(f, g)\in \mathscr{D}'(\mathbf{R})^2$ (“impossibility of the multiplication of distributions”).

d) Give examples of sequences $(f_n)$ and $(g_n)$ of test functions on $\mathbf{R}$ such that $f_n\rightarrow \varepsilon_0$ and $g_n\rightarrow \varepsilon_0$ in $\mathscr{D}'(\mathbf{R})$, and such that $f_n^2$ and $g^2_n$ converge in $\mathscr{D}'(\mathbf{R})$ to different limits.