Definition:Test Function Space

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Definition

Denote with $\map \DD {\R^d}$ the set:

$\set { \phi: \R^d \to \C, \text{$\phi$ is a test function} }$

of all test functions $\phi: \R^d \to \C$.

Then $\map \DD {\R^d}$ is called the space of test functions.

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$\map \DD {\R^d}$ is a vector space over $\C$, i.e.:

$\forall \psi, \phi \in \map \DD {\R^d} : \phi + \psi \in \map \DD {\R^d}$.

$\forall \phi \in \map \DD {\R^d} : \forall \alpha \in \C : \alpha \cdot \phi \in \map \DD {\R^d}$.

$\map \DD {\R^d}$ also has a topology.

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