Convergent Sequence in Test Function Space multiplied by Smooth Function

Theorem

Let $\alpha \in \map {C^\infty} {\R^d}$ be a smooth real multivariable function.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence in the test function space $\map \DD {\R^d}$.

Suppose $\sequence {\phi_n}_{n \mathop \in \N}$ converges in the test function space:

$\phi_n \stackrel \DD {\longrightarrow} \phi$

Then $\sequence {\alpha \phi_n}_{n \mathop \in \N}$ converges in the test function space.

$\alpha \phi_n \stackrel \DD {\longrightarrow} \alpha \phi$

Proof

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Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.4$: A glimpse of distribution theory. Multiplication by $C^\infty$ functions

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