Definition:Limit of Sequence/Test Function Space

Definition

Let $\map \DD {\R^d}$ be the test function space.

Let $\phi \in \map \DD {\R^d}$ be a test function.

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

Let $\sequence {\phi_n}_{n \mathop \in \N}$ converge to $\phi$ in $\map \DD {\R^d}$.

Then $\phi$ is a limit of $\sequence {\phi_n}_{n \mathop \in \N}$ in $\map \DD {\R^d}$ as $n$ tends to infinity which is usually written:

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

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples