Banach Spaces

Definition

A normed space $(X,\Vert \cdot \Vert )$ is a Banach space if it is complete as a metric space.

e.g. Let $\Omega \subset {\mathbb{R}}^n$ be a domain and $\alpha \in (0,1]$. A function $f:\Omega \to \mathbb{R}$ is said to belong to the Hölder space $C^{0,\alpha }(\Omega )$ if $\Vert f{\Vert }_{C^{0,\alpha }(\Omega )}:={\sup }_{x\in \Omega }|f(x)|+{\sup }_{\begin{array}{c}x,y\in \Omega \\ x\ne y\end{array}}\frac{|f(x)-f(y)|}{|x-y{|}^{\alpha }}<\infty .$

• The first term ${\sup }_{x\in \Omega }|f(x)|$ measures the boundedness of $f$.
• The second term ${\sup }_{x\ne y}\frac{|f(x)-f(y)|}{|x-y{|}^{\alpha }}$ measures the $\alpha$-Hölder continuity of $f$.

Equipped with this norm, $C^{0,\alpha }(\Omega )$ is a Banach space.