Editing Distribution (mathematics) (section)

====Expressing tempered distributions as sums of derivatives====

If <math>T \in \mathcal{S}'(\R^n)</math> is a tempered distribution, then there exists a constant <math>C > 0,</math> and positive integers <math>M</math> and <math>N</math> such that for all [[Schwartz function]]s <math>\phi \in \mathcal{S}(\R^n)</math>
<math display=block>\langle T, \phi \rangle \le C\sum\nolimits_{|\alpha|\le N, |\beta|\le M}\sup_{x \in \R^n} \left|x^\alpha \partial^\beta \phi(x) \right|=C\sum\nolimits_{|\alpha|\le N, |\beta|\le M} p_{\alpha, \beta}(\phi).</math>

This estimate, along with some techniques from [[functional analysis]], can be used to show that there is a continuous slowly increasing function <math>F</math> and a multi-index <math>\alpha</math> such that
<math display=block>T = \partial^\alpha F.</math>