Test-function space - Knowlpedia

For an open set $\Omega\subseteq\mathbb R^n$ , the test-function space is

\mathcal D(\Omega)=C_c^\infty(\Omega),

the vector space of smooth functions whose supports are compact subsets of $\Omega$ , equipped with its canonical LF topology. Choose compact sets $K_j\subseteq\operatorname{int}K_{j+1}$ exhausting $\Omega$ , and let $\mathcal D_{K_j}(\Omega)$ consist of functions supported in $K_j$ , with the seminorms

p_m(\varphi)= \max_{|\alpha|\leq m}\sup_{x\in K_j} \left|\partial^\alpha\varphi(x)\right|.

Then $\mathcal D(\Omega)$ carries the locally convex inductive-limit topology of the spaces $\mathcal D_{K_j}(\Omega)$ . This topology, not only the underlying set, is part of the definition.

Convergence

A sequence $\varphi_\nu$ converges to $0$ in $\mathcal D(\Omega)$ exactly when its supports are eventually contained in one fixed compact $K\Subset\Omega$ and every partial derivative converges uniformly to $0$ there. Supports that drift toward infinity or toward the boundary do not converge in the test-function topology merely because the functions and their derivatives converge pointwise.

Continuous functionals

A linear functional on $\mathcal D(\Omega)$ is continuous exactly when its restriction to each fixed-support space $\mathcal D_K(\Omega)$ is continuous. The resulting topological dual is the space of distributions on $\Omega$ . This is why distribution theory must specify the LF topology: algebraic linear functionals on $C_c^\infty(\Omega)$ are far more numerous.

Comparison with Schwartz space

When $\Omega=\mathbb R^n$ , every test function is a Schwartz function, and $\mathcal D(\mathbb R^n)$ is dense in $\mathcal S(\mathbb R^n)$ . The two spaces are nevertheless different: test functions have compact support, whereas Schwartz functions may have full support but must decay rapidly together with all derivatives.

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