Closed ball in a normed space

Question

Let $V$ be a normed space and denote by $$ B_1 := \{ v \in V: ||v|| \leq 1\}, \quad B_c := \{ v \in V : ||v|| \leq c \}, \quad cB= \{ c v : v \in B_1 \}, \quad c \geq 0. $$ Now clearly $cB \subset B_c$ for all $c \geq 0$. What are some examples where the converse inclusion does not hold?

Answer 1

There are none. If $v\in B_c$, then $v=c\frac vc\in cB_1$.

Answer 2

Always they are equal: $cB_1=\{ cv|v\in B_1\}=\{ cv:\ \lVert v\rVert \leq 1 \}=\{ cv: \ \lVert cv\rVert\leq c\}=B_c $