In a linear normed space, does the continuity of addition of the distance (metric) function imply that distance function is (shift-) invariant

Question

Suppose in a linear normed space $\mathcal{H}$, we have a distance function $\rho$, such that (1) $\rho(x,y)=0$ iff $x=y$, and $\rho(x,y)\geq 0$; (2) $\rho(x,y)=\rho(y,x)$; (3) $ \rho(x,z)\leq \rho(x,y)+\rho(y,z)$.

If the $\rho$ satisfies the continuity of addition, i.e. $$ \rho(x_n,x)\rightarrow 0~~ \&~~ \rho(y_n,y)\rightarrow 0 \Rightarrow \rho(x_n+y_n, x+y)\rightarrow 0$$

can we say this condition implies that this metric is invariant, i.e. $$\rho(x+z,y+z)=\rho(x,y), \forall x,y,z\in \mathcal{H} $$ ??

It feels like this should be true, but I am not so sure how to formally prove it. Thank you~~

Answer 1

This is false: define on $\mathbb R^2$, $d(x,y)=\vert x-y\vert $ if $x=cy\ $ for some $c\in \mathbb R$ and $d(x,y)=\vert x\vert +\vert y\vert\ $ otherwise.

Answer 2

I cannot see that the norm on $\mathcal{H}$ plays any role in the question, so I shall ignore it.

I take $\mathcal{H}$ to be a real or complex vector space provided with a metric, $\rho: \mathcal{H} \times \mathcal{H} \to \mathbb{R}$, with respect to which the binary operation of addition on $\mathcal{H}$ is continuous.

Although not much can be said on that basis alone, it might nevertheless be of interest to note that, if also the additive inverse operation $\mathcal{H} \to \mathcal{H}$, $x \mapsto -x$ is assumed to be continuous with respect to $\rho$ (this will be true, in particular, if the operation of scalar multiplication, $\mathbb{R} \times \mathcal{H} \to \mathcal{H}$ or $\mathbb{C} \times \mathcal{H} \to \mathcal{H}$, is assumed to be continuous), then it follows from a metrisation theorem for topological groups, proved independently by G. Birkhoff and S. Kakutani in 1936, that $\rho$ is equivalent to a shift-invariant metric.

There is a proof on Terry Tao's blog, but I haven't read that; the version I worked through - so long ago that I've forgotten the details, but I found it very clear - is in J. Dieudonne, Treatise on Analysis, Vol. II (1970), p.39f. (There is also a proof in J. L. Kelley, General Topology (1975), p.185f., but I can't remember if I read through it.)