Just something encountered during research and not sure every condition will be useful.
Let $(E, d)$ be a linear separable complete metric space (lsms) over the reals, that is, a linear space where the scalar multiplication and addition operations are continuous with respect to the metric.
Is there an upper bound $C<\infty$ such that $\frac{d(\theta x_1,\theta x_2)}{\theta d(x_1,x_2)}\leq C$ for any $\theta>0$, $x_1,x_2\in E$? This is true if $(E,d)$ is a Banach space, because $C=1$ in that case. I want to know if this is still true for general metric space.
One quick thing I think is true is that there exists $C_{1,\theta}, C_{2,\theta}\in (0,\infty)$ such that $C_{1,\theta}\leq\frac{d(\theta x_1,\theta x_2)}{\theta d(x_1,x_2)}\leq C_{2,\theta}$, because both $d(\theta x_1,\theta x_2)$ and $\theta d(x_1,x_2)$ are valid metric inducing the same topology on $E$. However, those two constants are related to $\theta$, which is not what I want.
Hope some people saw this before and could help me solving it. Thanks a lot!