3.1 Locally constant functions and distributions on a td space
Let \(X\) be a td space (i.e. totally disconnected, locally compact, Hausdorff space).
We write \(C^{\infty }(X)\) for the space of complex-valued locally constant functions on \(X\).
We write \(C_c^{\infty }(X)\) for locally constant functions with compact support.
A distribution on \(X\) is a \(\mathbb {C}\)-linear functional on \(C_c^{\infty }(X)\).
Distributions are additive and \(\mathbb {C}\)-linear.
3.1.1 Extension by zero and restriction
Given a locally constant function on a subset \(Z \subseteq X\), we extend it to \(X\) by zero outside \(Z\).
If \(Z\) is clopen, the extension by zero is locally constant.
If \(Z\) is closed and compact, the extension by zero has compact support.
A distribution on \(X\) restricts to a subspace \(Z\) by evaluating on the extension by zero of test functions on \(Z\).
The support of a distribution \(F\) consists of those points \(x\) such that every neighborhood of \(x\) contains a test function on which \(F\) does not vanish.
The support of a distribution is a closed subspace of \(X\).
If the support of a test function is disjoint from the support of \(F\), then \(F\) evaluates to zero on that function.