Can we visualize the closed balls for the space $l^{\infty}$ equipped with the $\sup$ norm

Question

The closed unit balls for the $l^{p}$ in $\mathbb{R}^2$ look like

I want to know could we also visualize the closed balls for the space $l^{\infty}$ equipped with the $\sup$ norm .

Thanks

Answer 1

For any $n$, the set $\{(x_1,\dots,x_n):\sup|x_i|\le 1\}$ is the $n$-dimensional hypercube of sidelength $2$, namely $[-1,1]^n$.

Normally, one does not directly visualize an infinite-dimensional object, but rather infer its geometric properties from a finite-dimensional model. For example: the unit cube of $\ell^\infty$ is not strictly convex (its boundary contains some line segments), has a non-smooth boundary, and infinitely many "vertices" (extreme points) which are the sequences of $\pm 1$.