Past exam of the differential equations and applications course of the University of Oxford / 2010 / A / 1 / ii

A distribution is a continuous linear functional on the space of test functions $\mathcal{D}(\mathbb{R})=C_c^{\infty }(\mathbb{R})$. A precise useful form of continuity is: for every compact set $K\subset \mathbb{R}$ there exist an integer $m\ge 0$ and a constant $C_K$ such thatEquivalently, if a sequence of test functions has supports in one fixed compact set and every derivative tends uniformly to zero, then its images under $T$ tend to zero. Keeping a common compact support is part of the test function topology; mere pointwise convergence is insufficient.

Linearity follows from linearity of differentiation and point evaluation. For a test function supported in a fixed compact set $K$, its value and derivative vanish at any point outside $K$. ThereforeThis proves the continuity estimate with $m=1$ and $C_K=2$ for every $K$. Thus the stated mapping is a distribution of order at most one, supported in $\{0,1\}$. In particular, a sequence tending to zero in the test function topology has both ${\varphi }_n^{\prime }(0)\to 0$ and ${\varphi }_n(1)\to 0$.

The distributional derivative is defined by moving the derivative onto a test function and changing the sign:Since $⟨{\delta }_a,\varphi ⟩=\varphi (a)$, one has $⟨{\delta }_0^{\prime },\varphi ⟩=-{\varphi }^{\prime }(0)$. HenceThe minus sign in front of ${\delta }_0^{\prime }$ is essential: it makes its action the positive derivative evaluation required here.