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How to add and multiply on fractional vector space

Please, answer in layman terms. I don’t understand the notation of Supersimetry and Super vector spaces.

If a fractional “vector space” (or his fractional equivalent) has dimension between 1 and 2: $1 \,\leq\, (d=1+\frac{1}{n}) \,\leq\, 2 \;\;\;\; n\in \mathbb{N}$, and a generic vector is $(r,f)$ with $r \in R$ and $f \in \mathbb{R}$, being r the real part, and f the fractional one:

How are the vectors added? What is the formula? Is $(r_1,f_1)+(r_2,f_2)=(r_1+r_2,f_1+f_2) $?

How are those vectors multiplied? Like a complex number? What is the fractionary imaginary unit $\omega$?

That would make them the same as a complex number, so I guess that's wrong.

I guess that the fractional number is redundant, like it is periodic.

Can this be $r*e^{\omega. \theta}=r(cos(\Omega.\theta)+\omega.sin(\Omega. \theta))$

so, $\Omega$ would change the periodicity of the number, so the same number will be repeated with faster frequency than in complex numbers. Is that appropriate? what would be $\Omega$ as function of $1+\frac{1}{n}$?

I know that the question is vague. I do not really know exactly how to ask it.

I imagine that numbers with a fractional dimension $1 \leq \alpha \leq 2$ belong to a space which is a surface whose area grows with a non integer power of the radius. Like complex belongs to a plane whose area grows with the square of the radius.

depi zixuri
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    I have never heard of a "fractional" vector space, nor of a "vectorial space" if this is a different concept. Please provide a reference citation where you found this terminology. – hardmath Feb 22 '16 at 17:22
  • Sorry, I mean "vector space". I made a literal translation from my language. Is edited. – depi zixuri Feb 22 '16 at 17:37
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    I do not really know if fractional spaces can be vector spaces. I'm looking for the definition of a number with a fractional dimension, like something between reals and complex spaces. – depi zixuri Feb 22 '16 at 17:39
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    A similar question is here, but I do not understand the symbology in the answer. http://math.stackexchange.com/questions/1466820/vector-spaces-with-fractional-dimension?rq=1 – depi zixuri Feb 22 '16 at 17:40
  • A vector space is defined to have vector addition and scalar multiplication, where scalars are numbers in a field. So the real numbers $\mathbb R$ are a field and the complex numbers $\mathbb C$ are a field. Dimension is a word used to mean many different things in math, a few of which allow for a fractional value. But dimension of a vector space with respect to its field of scalars is defined to be a cardinal number, and this does not admit a fractional value. – hardmath Feb 22 '16 at 18:42
  • You have made some significant clarifications in your comments, and I thank you for that. However it would improve your Question if you incorporated these into Edits of the Question. I have tried to answer the Question as your Comments have changed its sense, and I hope you will feel free to ask for clarification if it would help. – hardmath Feb 22 '16 at 19:29

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If we concern ourselves with number systems rather than vector spaces, then potentially some different notion of dimension can be defined. However (and this is already argued somewhat by the OP) there is no number system between the real numbers and the complex numbers with a definition of addition and multiplication operations consistent with their definitions on the real numbers. So the simplest possible example of number systems of "fractional dimension" you envision does not exist.

See Haudorff-Besicovitch dimension for a definition that allows fractional values, but this is for metric topological spaces and not for "number systems" generally. However "Hausdorff dimension" (as it is usually called) does generalize the notion of dimension of a vector space, since when we apply it to a (finite) $n$-dimensional real vector space with the product topology induced by the usual metric topology on the real numbers, we get the same number $n$ for the Hausdorff dimension.

As far as explaining this in layman's terms, the discussion seems to require some degree of mathematical preparation in order to get "fractional dimensions". It is true that the concept of fractals has penetrated into popular culture somewhat, mainly in the form of attractive pictures, but this hardly constitutes an explanation in mathematical terms.

hardmath
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