Banach space

A Banach space is a normed vector space $(X,\|\cdot\|)$ such that every Cauchy sequence in $X$ converges (in the norm) to a point of $X$.

Equivalently, the metric $d(x,y)=\|x-y\|$ makes $X$ a complete metric space . Completeness is a property of the metric induced by the norm , and it is essential for many limit processes in analysis.

Examples:

• $\mathbb{R}^n$ with the Euclidean norm (indeed, $\mathbb{R}^n$ is Banach for any norm).
• The space $C([0,1])$ of continuous real-valued functions on $[0,1]$ with the sup norm $\|f\|_\infty=\sup_{x\in[0,1]}|f(x)|$.