# An inner-product space or an inner product space?

I have seen both of these being used for referring to a mathematical object (a vector space with a certain structure on it called the inner product). I am assuming both are accepted since they appear in resources. Is one of them better than the other from a linguistic point of view?

To me, the one with the hyphen seems tidy, and I think we should use the hyphen as this is a compound word.

• You're likely to be told to ask this on Mathematics Stack Exchange. That said, I agree with your preference - the defining characteristic of the space is the inner product. The hyphen helps resolve a possible ambiguitiy. Commented Apr 12, 2023 at 13:48
• "inner product space" is fine and "pre-Hilbert space" is a synonym. Commented Apr 12, 2023 at 14:06
• Do you mean to capitalise the first one and not the second, as in the title? Commented Apr 12, 2023 at 14:38
• @StuartF Thank you for pointing this out; I just fixed it. It's a typo.
– user205527
Commented Apr 12, 2023 at 16:54

Although there exists a tendency in cases of premodification by a term made up of two words to hyphenate them (ex.: job-aptitude test ), the dominant form is "inner product space".

Nevertheless, you might follow the rare users of "inner-product space" (3 cases, inner product space) ; the particular choice made is not considered to be a real error.

For mathematicians and most of the time for students it matters little because they know that "inner" does not modify "product space". Before speaking of inner-product (vector) spaces they had to study vector spaces and then, later, inner products. For the non-initiated it is important as far as it makes clear that there exists an entity called "inner product" and that it modifies "space".

In the case of hypothetical developments in set theory, if mathematicians came to find useful that a product space is "inner" (whatever that would mean in this context), then the use of "inner product-space" would be possible without ambiguity, at least on paper.

We don’t hyphenate, for example, high school in high school reunion, and neither should we hyphenate inner product in inner product space

Says The Chicago Manual of Style:

It wouldn’t be incorrect to write “middle- and high-school students.” But both “middle school” and “high school” are listed in Merriam-Webster as unhyphenated noun phrases; when they are used attributively, they can remain unhyphenated.

In general, any compound that’s rarely hyphenated in real life can remain unhyphenated as a phrasal adjective if the meaning remains clear without the hyphen. This goes double for any compound that’s listed in a dictionary without the hyphen. So write “middle and high school students.”
Source: The Chicago Manual of Style Q&A

By way of going double, here’s inner product under inner in the OED:

A. 1. k. inner product n. [translating German inneres Produkt (H. Grassmann Die lineale Ausdehnungslehre (1844) p. xi): so named because an inner product of two vectors is zero unless one has a component ‘within’ the other, i.e. in its direction] Mathematics the sum of the products of corresponding components of two real vectors . . .
Source: Oxford English Dictionary (login required)

So, no hyphen is needed for this noun phrase in an attributive position.