Urysohn's universal metric space $\Bbb U$ satisfies the following surprising property:
Whenever $i\colon\Bbb U\to E$ is an isometric embedding into a normed vector space such that $0\not\in i(\Bbb U)$, then $i(\Bbb U)$ is linearly independent.
Call a metric space satisfying the property above a linearly independent space (I don't know if there is already a name for them, I just made this one up).
I'm interested in examples and properties of linearly independent spaces, to begin with are there nontrivial, meaning not singletons, examples that are "fundamentally" different from $\Bbb U$? Must those spaces satisfy some kind of universality and/or homogeneity property? Do they have other interesting properties?
For example a simple observation is that linearly independent spaces have infinite covering dimension unless they are finite, in which case they must be a singleton, but I haven't been able to say much more about them.
