Well-defined $\xi$-weighted (Euclidean) norm

Question

Suppose $\xi$ is a vector, that is used for $\parallel z\parallel_\xi$ calculation. Should every element of $\xi$ be positive, $\xi(i)>0$?

Answer 1

The answer depends on how you define the weighted norm. For $$\|z\|_\xi = \left(\sum_i z_i^2 \xi_i\right)^{1/2}$$ Then yes, $\xi_i>0$ is required for this to be a true norm. If instead you define it as $$\|z\|_\xi = \left(\sum_i (\xi_i z_i)^2\right)^{1/2}$$ then the condition is simply $\xi_i\neq 0$.