If $\|x\|=1$ just means the vector $x$ has length one – Then why is the unit sphere defined as $S=\{x\in X| \quad \|x\|=1\}$?
let $X$ be a normed linear space with the Euclidean norm, then letting the vectors have their beginning at the origin works, but if $\|x\|=1$ just means the vectors lie anywhere in space and have length one, then I can't see why this gives a sphere.
Best Answer
It is a generalization of the spheres we already know.
Consider in $\Bbb R^3$ the norm given by $\|{\bf x}\|^2 = \sum_{i=1}^3x_i^2$. Then the set $S = \{ {\bf x} \in \Bbb R^3 \mid \|{\bf x}\| = 1 \}$ is a sphere centered in the origin with radius $1$. In $\Bbb R^2$, we have that $S$ is a circle centered in the origin with radius $1$. If you consider in $\Bbb R^2$ the norm $\|{\bf x}\| = \max\{|x_1|,|x_2|\}$, the set $S$ is a square centered in the origin, with sides parallel to the coordinate axes and side $2$. And so on.