Unit sphere of a normed space

For a normed vector space $E$, its unit sphere is

It is the boundary of the closed unit ball $\{x:\lVert x\rVert\le1\}$. The term “sphere” refers to the given norm: unless the norm comes from an inner product, $S_E$ need not be round in Euclidean coordinates.

Every nonzero vector is a positive scalar multiple of a unique point of $S_E$, namely $x/\lVert x\rVert$.