Banach space

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.

By contrast, the polynomials on $[0,1]$ with the sup norm are not a Banach space: uniform limits of polynomials can be arbitrary continuous functions, so the polynomial subspace is dense but incomplete.

• $\mathbb{R}^n$ 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 $\|f\|_\infty=\sup_{x\in[0,1]}|f(x)|$.