I have seen the expression, "X is the Hilbert space in which the element x lives". As a native speaker, this seems quite sloppy to me. Is there a more succinct way to formulate this expression?
This usage of lives is an anthropomorphisation: human attributes and behaviours are affixed to an abstract object in order to make it more accessible to the usual human thought patterns.
If you want to, you can substitute occurs:
X is the Hilbert space in which element x occurs.
or, since lives implies it cannot "live" elsewhere (which is why anthropomorphizations are such a powerful tool)
Element x only occurs in Hilbert space X.
You ask about succinctness, but then you put quotes around "lives," so it's not clear what exactly you're asking.
To make it more succinct, the word "where" can be substituted for "in which." That would make it more succinct. While using "where" to mean "in which" for the abstract is widely considered to be informal and is something one would avoid in an academic paper, like a thesis or dissertation, that isn't the case here. That's because the sentence use the word "space," making the meaning literal rather than abstract.
Regarding "lives," that's about as succinct as succinct gets.
I'm wondering if you're suggesting that another verb might be more apt, like maybe because the word "element" refers to something that is inanimate. I can't be sure that's what you mean, though, because "element" can refer to living things, too, like when we say, "An unsavory element has moved in down the street and has taken up living in Mr. and Mrs. Smith's old house." Without more context, for the singular contextual clue of "Hilbert" means nothing to me, I can't be sure what you're driving at.
What I can say about using "lives" there, though, is I don't have a problem with using the verb "lives" for the subject "element" even if "element" refers to something inanimate. Such a usage would merely employ imagery, an imagery that paints a very vivid picture that could be quite apropos to the circumstances, even most succinct.
While you could certainly switch "lives" out for another verb like "exists" or "occurs," doing so would come across quite differently to readers because using the verb "lives" for the subject "element," if inanimate, personifies or anthropomorphizes "element" such that it imbues it with a certain unrelenting permanence, like as if it willed itself there, or imbues it with a certain unpredictable, ongoing effect on the environs, like as if it behaved with freewill.
Therefore, because of the substantive difference in how "lives" comes across and because I operate from a position that people say what they mean and mean what they say unless I'm on solid ground to know otherwise, I would be extremely reluctant if I were you to say that "lives" is wrong or to criticize another writer as being "sloppy" or less "succinct" for using "lives" there.
I don't see a problem with "lives", said of imaginary inhabitants of various dimensional spaces. Many mathematicians and scientists have read Abbot's Flatland, I imagine, and found it an aid to the intuition to think of what it would be like to live in a space with a different dimensionality.
A Hilbert space is by definition an inner product space. Even simpler, it's a vector space with an operation that satisfies certain things. A metric space, in turn, is just a set satisfying certain (other) things. In the end, what we have is a set (e.g. an infinite set of points) on which some operations are defined with certain criteria (e.g. a distance and an inner product).
So when someone says a point x lives in a Hilbert space X, what they mean is that X is a set of points with certain associated operations and x is an element in that set of points.
As such, a succinct way of expressing that is:
Let X be a Hilbert space with x in X.
Indeed, as others have noted, the turn of phrase in question seems to be an attempt at making the whole thing more lively. This isn't unheard of, and according to MathWorld, Hilbert spaces can even be a punchline (well, I assume you need the right audience):
A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005).