There is a theorem stating that every metric space embeds isometrically into $\ell _{\infty}$.
My question: is there a generalized result for pseudo metric spaces embedding isometrically into semi-normed spaces? Is/would this result be interesting/useful?
Thank you very much for taking the time to answer my question.
Yes. If $(X, d)$ is a pseudometric space, then we may may define an equivalence relation $\sim$ on $X$ by $x \sim y$ iff $d(x, y) = 0$. Then $(X/{\sim}, d’)$ with $d’([x], [y]) = d(x, y)$ is a metric space, so there is some isometric embedding $\phi: X/{\sim} \to \ell_\infty(I)$. Now, let $\kappa = \sup_{x \in X} |[x]|$ and pick a vector space $V$ of cardinality at least $\kappa$ (say, $V = \mathbb{K}[\kappa]$). For each equivalence class $[x]$, fix an injection $\psi_{[x]}: [x] \to V$. Now, equip $\ell_\infty(I) \times V$ with the seminorm:
$$p(f, v) = \|f\|_{\ell_\infty(I)}$$
Then $\pi: X \to (\ell_\infty(I) \times V, p)$ defined by,
$$\pi(x) = (\phi([x]), \psi_{[x]}(x))$$
is an isometric embedding. (I don’t see how such a result is in any way interesting or useful though.)
