Editing Distribution (mathematics) (section)

====Topology on ''C''<sup>''k''</sup>(''K'')====

As before, fix <math>k \in \{0, 1, 2, \ldots, \infty\}.</math> Recall that if <math>K</math> is any compact subset of <math>U</math> then <math>C^k(K) \subseteq C^k(U).</math>

{{block indent|em=1.5|text='''Assumption''': For any compact subset <math>K \subseteq U,</math> we will henceforth assume that <math>C^k(K)</math> is endowed with the [[subspace topology]] it inherits from the [[Fréchet space]] <math>C^k(U).</math>}}

If <math>k</math> is finite then <math>C^k(K)</math> is a [[Banach space]]{{sfn|Trèves|2006|pp=131-134}} with a topology that can be defined by the [[Norm (mathematics)|norm]]
<math display=block>r_K(f) := \sup_{|p|<k} \left( \sup_{x_0 \in K} \left|\partial^p f(x_0)\right| \right).</math>