Let $\mathbb K\in\{\mathbb R,\mathbb C\}$, let $\mathbb N:=\{1,2,3,\dots\}$, and let $(X,\|\cdot\|_X)$ be an infinite-dimensional [normed vector space](/page/Normed%20Vector%20Space) over $\mathbb K$. Define the closed unit ball by $\overline{B}_X(0,1):=\{u\in X:\|u\|_X\le 1\}$. Define the norm metric $d_X:X\times X\to\mathbb R$ by $d_X(u,v):=\|u-v\|_X$. Then $\overline{B}_X(0,1)$ is not compact in the norm topology induced by $d_X$.