Fractional Dimensional Spaces and Fourier's Trick

One doesn't usually think of the existence of the standard basis for euclidean vector spaces as related to fourier series. It is easy to show that there is a rather direct correspondence. If we take a slightly non standard method of providing a coordinate system to the points in Rⁿ we let every point p be labeled with a the function defined using the usual dot product as f_p(x) = p * x where * is the vector dot product and x is any other point in Rⁿ Now to obtain the dot product of two points we express each point as a linear combination of the standard basis vectors and then take the sum of the products of the corresponding coefficients in the standard basis expansion. More to the point if you take the dot product of one of the basis vectors and any point you obtain the coordinate of that point with respect to that basis vector. Thus the dot product of two points is the sum over the standard basis of the products of the dot products of the two points under consideration and the i'th vector in the standard basis.

Now consider what happens when we take our above functions mapped to points. We can obtain the dot product of the two points as the integral of the product of the functions associated to the points over the unit sphere of vectors. If the space under consideration is integer dimensional space then relatively simple exercises in linear algebra show that any point can be expressed as the linear combination of d linearly independent vectors (d being the dimension of the space). So (as is kind of obvious) if a (finite) standard basis exists then the dimension of the space is integer.

This is all based on the fact that the space has a vector space structure set up on it. I posit that vector space structure does not in itself imply the dimension of the space to be of integer dimension. If this is true then it means that any non-integer dimensional space would have to have an infinite basis