Definition

For $f:X\to \mathbb{C}$ essential bounded we define the infinity Norm as $$\|f\|_{\infty}:=\inf\{ M\ge 0:M\text{ is an essential bound for }f \}$$ ## Examples The $\infty$-Norm, denoted as $\|\cdot\|_{\infty}$, for any arbitrary $v\in V$ is defined as follows: $$\begin{align*} \mathbb{F}^{n}&: \ \|v\|_{\infty}=\max\{|v_{i}|:i\in\{1,\ldots,n\}\}\\ \mathbb{F}^{\infty}&:\|(v_{i})_{i\in\mathbb{N}}\|_{\infty}=\sup\{|v_{i}|:i\in\mathbb{N}\}\\ C^{0}([a,b];\mathbb{F})&:\|f\|_{\infty}=\sup\{|f(x)|:x\in[a,b]\} \end{align*}$$