Let $C_c^\infty(\mathbf{R}^d)$ be the family of all compactly supported smooth functions equipped with the test function topology. Let $C^\infty_{\text{loc}}(\mathbf{R}^d)$ be the class of all smooth functions, equipped with the topology of locally uniform convergence of the function and all it's derivatives.

It is stated in Treves, Chapter 41 that the multiplication operator

$$ C^\infty_c(\mathbf{R}^d) \times C^\infty_{\text{loc}}(\mathbf{R}^d) \to C_c^\infty(\mathbf{R}^d) $$

is not continuous. It is certainly separately continuous, and sequentially continuous in each variable. But I can't find a resource that proves that the map is not continuous. Why is this the case?



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