It is well-known that you can take a normed space and produce a metric on it given by $d(x, y) = |x - y|$. This works well for things like Euclidean space. However, I have seen people (notably people working with relativity) working with a "norm" that is nonpositive, that is, where you have vectors with $|v|^2 < 0$. Using these "norms" stops the above creation of a metric space from working. When I ask people how you can still do analysis on these spaces, they say to define the norm while pretending the space is Euclidean.
To make this idea more rigorous, I came up with this idea: Let $V$ be a vector space over $\mathbb R$ and $Q : V \to \mathbb R$ be a quadratic form. If we have another positive-definite quadratic form $Q^+ : V \to \mathbb R$ such that $Q(v) \leq Q^+(v)$ for all $v \in V$, then we can define a metric on $V$ by using $Q^+$ as above.
This leads me to the following two questions:
- Is this sufficient to do analysis on these spaces? Will knowing that $Q(v) \leq Q^+(v)$ be enough to do anything we might want to do with $Q$ and operations related to it, such as its associated symmetric bilinear form?
- Does such a $Q^+$ always exist? The countably-infinite-dimensional case is easy through orthonormalization, but I'm not sure about the uncountably-infinite-dimensional case. I know that there exist uncountably-infinite-dimensional spaces with no orthogonal basis, so I'm slightly worried that there is a counterexample to this conjecture.