Reason for the term "smooth" 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?
 A: 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.
A: Let us consider the space $X=\mathbb R^2$ with the $\ell^p$-norm.
Then for $p=1$ or $p=\infty$ we can see that the unit ball has kinks, and does not look smooth.
It can be shown, that there are points $x\in X$ such that there is more than one functional $f$
with $\|f\|=1$ and $f(x)=\|x\|=1$.
(For example, if $p=1$ consider the point $x=(1,0)$, then $g(x)=x_1$ and $h(x)=x_1+x_2$ are possible choices for $f$.)
For $p$ with $1<p<\infty$ the unit ball looks smooth (its boundary is differentiable).
And it is possible to show that for each $x\in X$ there is exactly one functional $f$
with $\|f\|=f(x)=\|x\|=1$.
I hope this is sufficient motivation for the term "smooth" for a normed space.
