Let $(V,+,\cdot,\|.\|)$ be a normed vector space. Can we reconstruct addition $+$ of vectors and scalar multiplication $\cdot$ if we are given only the underlying set $V$ and the norm $\|\cdot\|\colon V\to\Bbb R$?
Clearly, we can find $0\in V$ as it is the only element of norm $0$, and we know $1\cdot v=v$ and $0\cdot v=0$. And we have the topology. But is that enough to reconstruct the missing operations?
Best Answer
No. If this were true, it would mean that every norm-preserving self bijection of a normed vector space preserved addition and scalar multiplication. A counterexample is $V=\mathbb{R}$ with $\|\cdot\|$ given by the usual absolute value. Then the self bijection $$ \varphi(x)=\begin{cases}x&:|x|\leq 1,\\ -x&:|x|>1\end{cases} $$ preserves the norm, but not the vector space structure.