Let $(V, \|\ \cdot\ \|)$ be a normed space over $\mathbb{R}$ or $\mathbb{C}$ and let $W\subset V$ be a linear subspace. We define the following equivalence relation on $W$: $$x\sim y\iff x-y\in W.$$ Then, we can define the quotient space $$V/W=V/\sim=\{[x]:x\in V\},\ \ \text{where}\ \ [x]\ \ \text{is the equivalence class}.$$ And we define the quotient "norm" to be $$\|[x]\|_{V/W}=\inf_{y\in W}\|x-y\|.$$
It is known that $\|\ \cdot\ \|_{V/W}$ is a semi-norm, and when $W$ is not closed, it may fail to be a norm since it may fail the property that $$\|[x]\|_{V/W}=0\iff [x]=[0].$$
One counterexample has been given here: Concrete counterexample for norms on quotient spaces, but it seems that the definition of the quotient in this post is a little bit different than mine.
Therefore, I tried to come up with an example by considering $c_{00}$ as a subspace of $\ell^{\infty}$ (as long as the elements in $c_{00}$ comes from $\ell^{\infty}$, the subspace property is clear).
Firstly note that $c_{00}$ is not closed in $\ell^{\infty}$. Indeed, for each $n\in\mathbb{N}$, set $$x_{n}:=\Bigg(1, \dfrac{1}{2},\dfrac{1}{3},\cdots,\dfrac{1}{n},0,0,\cdots\Bigg)\ \ \text{and}\ \ x=\Bigg(1, \dfrac{1}{2}, \dfrac{1}{3},\cdots\Bigg).$$ Then $(x_{n})_{n=1}^{\infty}\subset c_{00}$ and $\|x-x_{n}\|_{\infty}=\frac{1}{n+1}\longrightarrow 0$, and thus $x_{n}\longrightarrow x$ in $\ell^{\infty}$, but $x\notin c_{00}$.
However, I cannot come up with a sequence $x\in c_{00}$ (so that $[x]=0$) but $\|[x]\|_{\ell^{\infty}/c_{00}}\neq 0.$
Any idea? Thank you!
Best Answer
You have $x \in \ell^\infty \setminus c_{00}$ (and consequently $[x] \neq 0$), so if $\|\cdot\|_{V/W}$ were a norm, then we would expect $\|[x]\|_{V/W} > 0$. However, $$0 \le \inf_{y \in c_{00}} \|x - y\| \le \|x - x_n\| \to 0,$$ as $n \to \infty$. By squeeze theorem, we have $$0 = \inf_{y \in c_{00}} \|x - y\| = \|[x]\|_{V/W},$$ proving $\|\cdot\|_{V/W}$ is not a norm after all.