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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?

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    You're probably venturing into the crossover between the English language and the language of Mathematics. So English language rules don't always apply. Mar 29, 2019 at 15:40
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    What's wrong with that is the house in which the woman lives? Would you rather it be phrased that is the house where the woman lives? If so, then rephrase your sentence the same way: X is the Hilbert space where the element x lives. Or are you objecting on different grounds? If so, what are they? Mar 29, 2019 at 16:17
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    This is informal, but I have heard almost the exact sentence in many academic talks and graduate level classes, so I'd call it verbally acceptable. I probably wouldn't put it in a paper though.
    – BenL
    Mar 29, 2019 at 21:50
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    Mathematician here. This is very common, to anthropomorphize mathematical objects when speaking/writing informally with fellow experts. I even recall a professor in graduate school who, when referring to various sets of vertices in a graph, said, "And then these guys fight with these guys". That would never make it into an academic paper, but it made sense at the time. Apr 4, 2019 at 22:15
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    If this is the language that mathematicians use then it is the language that mathematicians use. Some people object to anthropomorphising things but people do it all the time (that's why there is a word for it). "Exists" is probably a good alternative if you need something more formal.
    – user323578
    Apr 29, 2019 at 7:06

5 Answers 5

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You could say:

X is in the Hilbert space to which the element x belongs.

Though I don't think using "lives" is necessarily wrong (it is a bit informal, though).

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  • I agree that "lives" does not seem to be necessarily incorrect, but indeed, a tad informal. Additionally, I'd like to ask: do you think it's better to write "X is the Hilbert space in which the element x belongs" or your version? I suppose it depends on whether it's better to say an element "belongs to" or "belongs in" a space. In retrospect, I suppose your version seems more correct.
    – Kemal Raik
    Mar 29, 2019 at 15:28
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    @KemalRaik read this: ell.stackexchange.com/questions/32375/…. Summary: Use belongs to.
    – satnam
    Mar 29, 2019 at 20:35
  • Belongs has the connotation that it might be found outside of it and would need to be put back in.
    – Jim
    Mar 30, 2019 at 4:34
  • @Jim It largely depends on the context. A Hilbert space is a mathematical notion. Based my coursework, I believe "belongs" is an accurate term.
    – AleksandrH
    Mar 30, 2019 at 12:30
  • @AleksandrH I was really responding to the ‘belongs in’ question, but I didn’t make it very clear. I like ‘belongs to’ as well. But not ‘belongs in’- which has the connotation I mention.
    – Jim
    Mar 30, 2019 at 15:10
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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.

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  • This is far beyond the ESE, but can anything be expressed in only a single Hilbert space? If not, then perhaps "lives" is just wrong instead of informal. Perhaps "X is a Hilbert space in which element x can be expressed" would then be more accurate.
    – BenL
    Mar 29, 2019 at 22:14
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    I would use "exists" rather than "occurs". "Occurs" gives the impression that it pops into existence somehow (which may be true, but is a whole 'nother chapter in physics).
    – Hot Licks
    Mar 29, 2019 at 23:47
  • Another option is "resides". And maybe some synonyms of that: dwells, occupies, abides, et al.
    – Hot Licks
    Mar 29, 2019 at 23:49
  • I disagree that "lives" implies it can't live elsewhere. (I live in more than one country!) "Exists" is much better than occurs as we are talking about the element "being" in that space, not doing something in that space.
    – user323578
    Apr 29, 2019 at 7:03
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You ask about succinctness, but then you put quotes around "lives," so it's not clear what exactly you're asking.

Where

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.

Lives

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.

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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.

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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).

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