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I am trying to find out if there is a theory of vector spaces in non-integer dimension?

My reasoning is like this. One can form a space of vectors of the form $(x,y,z)\in \mathbb{R}^3$.

If one puts a restriction on this vector space like this $x+y+z=0$ then now $(x,y,z)\in \mathbb{R}^2$

Or indeed one can put a restriction $(x+y+z)^2+(2x+3y+4z)^2=0$ then $(x,y,z)\in \mathbb{R}$.

But can one put a restriction such that $(x,y,z)\in \mathbb{R}^d$ where $2<d<3$?

One thought I had was to define a function $f(x,y,z)=0$ where $f$ is zero only if the point $(x,y,z)$ is in a 3d version of the Sierpiński triangle. (Not entirely sure how one would define such a function). But it seems like one can define a non-integer vector space as a pair which consists of an integer vector space together with a function which restricts the values.

Now assuming this is a definition of a non-integer vector space, could there exist any analogues of things like the rotation group $SO(d)$ or Euclidean geometry, or any spheres are things from linear algebra?

One basic thing would be two define a matrix in non-integer dimensions, which would presumably consist of a matrix in an integer dimension together with some restriction on the values of the matrix as before.

zooby
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Vector spaces have a axiomatic definition. If your “vector space” is a vector space according to the mainstream definition, then it is either going to be finite “natural number” dimension as you are already aware, or an infinite-dimensional one.

Further Remarks:

I have presumed that you’re hoping for a non-integer dimension in the sense of the “number” of basis vectors one has for say $\mathbb{R^3}$. Nonetheless, that does not mean a notion of “dimension” could not be defined to accommodate non-integer values. While I won’t venture into that here, let me say that the “restriction” you made mention surely leads to “subsets” of the bigger space you begin with; indeed one can then ask what “subsets” they form and what “dimension” they have. If you require the subsets to still be a “vector space” as you know it to be they’d be of smaller integer dimension (or, even possibly, the same dimension); if they’re not vector spaces anymore —the easiest being that they’d be affine or convex—then we can ask for their relative dimensions in the bigger space as well. These will be integer dimensions.

Jack LeGrüß
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  • Well, true, but the definition of "number" has been extended many times. Also the definition of "multiplication" so, its not unheard of that a word or phrase in mathematics can be extended to include new things. Or that new concepts borrow words from older fields of mathematics. For example, "can we invent a system of 'numbers' that don't commute". Or lets call this combination of matrices "multiplication". – zooby Oct 11 '20 at 15:07
  • @zooby: why I do understand your logic (and it’s absolutely true that it lies in what “dimension” is); however, I believe the OP is about the algebraic dimension—as the cardinality of a basis—of the space over its underlying field. I’d add further remarks, for clarity. – Jack LeGrüß Oct 11 '20 at 15:18