TITLE: Reason for the term "smooth"
QUESTION [8 upvotes]: A normed space $X$ is said to be smooth if for $x \in X$ with $||x||=1$ there exists a unique bounded linear functional $f$ such that $||f||=1$ and $f(x)=||x||$. Why the term "smooth" comes?

REPLY [6 votes]: This condition implies that the function $F \colon x \mapsto \|x\|$ is (Gâteaux) differentiable in $x$ with $F'(x) = f$.
Indeed, the set
$$
\{ f\in X^* \mid \|f\| \le 1 \text{ and } f(x) = \|x\|\}$$
coincides with the convex subdifferential of $F$ at $x$.
If this subdifferential is a singleton, then $F$ is differentiable.