Realizing this covers a lot of territory, i.e. projective geometry, algebraic spaces, manifolds, etc., I'm basically looking for an exhaustive tabulation.

[edit, post comment]

I better understand part of why my original question was unclear: I mean space in the very generalized sense of the word, e.g. Lay Euclidean, vector, topological space (including metric), etc. Perhaps the word context would be better than the generic space.

The motivation is research towards having a complete survey of the overarching concept of projection in contexts such as geometry, algebra, and, e.g. functions. In many areas of human thought, confusion can occur when a term's meaning changes depending on the context it's used in. For instance, but not likely to be confused: function "space" is a different animal than Minkowski space. But there are areas where the difference requires either a deeper understanding of the subject, or the author takes care to clarify the difference. This is less of a problem for those versed, but can be a headache for students.

I'd like to ask for your expertise as I'm not confident that sources such as

http://mathworld.wolfram.com/ProjectiveSpace.html or https://en.wikipedia.org/wiki/Projective_space

are complete enough.

Is this clearer?

[Previously and unclear - and perhaps an example of the very problem! :]

  • P^n(R) = real projective space,
  • P^n(C) = complex " ",
  • (more generally) = vector spaces, (projective) Hilbert spaces, etc etc.

If you all don't think this is too broad, what TAGS should I use?

  • 5
    $\begingroup$ What's the motivation? What counts as different? I mean I can write $P^n (K)$ for any field $K$ are these all different for different fields? Likley not, but then it seems they are for the reals and the complex numbers. More than too broad, I'd worry it is unclear what you mean precisely. $\endgroup$
    – quid Mod
    Sep 20, 2016 at 18:49
  • $\begingroup$ @quid - hmmm, yes I see what you're saying, mention of P^n(K) is confusing, perhaps because it's too low level. I'll edit and hopefully clarify... THX $\endgroup$ Sep 21, 2016 at 16:11


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