Editing Smoothness (section)

===The space of ''C''<sup>''k''</sup> functions===
Let <math>D</math> be an open subset of the real line. The set of all <math>C^k</math> real-valued functions defined on <math>D</math> is a [[Fréchet space|Fréchet vector space]], with the countable family of [[seminorm]]s
<math display="block">p_{K, m}=\sup_{x\in K}\left|f^{(m)}(x)\right|</math>
where <math>K</math> varies over an increasing sequence of [[compact set]]s whose [[union (set theory)|union]] is <math>D</math>, and <math>m=0,1,\dots,k</math>.

The set of <math>C^\infty</math> functions over <math>D</math> also forms a Fréchet space. One uses the same seminorms as above, except that <math>m</math> is allowed to range over all non-negative integer values.

The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of [[partial differential equation]]s, it can sometimes be more fruitful to work instead with the [[Sobolev space]]s.