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Martin Sleziak
Martin Sleziak
7:05 AM
13
Q: Vector spaces with fractional dimension

preguntonCan the notion of vector space or algebra over a field be meaningfully extended to fractional dimensions, so that for example $\mathbb{R}^{-2/3}$ makes sense? Has this been explored somewhere? I know that super vector spaces can be thought of as one way of generalizing vector spaces to negative ...

vector-spaces dimension-theory
-1
Q: Is there this notation in mathematics $\mathbb{R^{\frac{1}{2}}}$ and is it meant the space dimension is $\frac{1}{2}$?

zeraoulia rafikI'm not familiar with algebra theory, but I'm interested to know if there is an ensemble in mathematics for which the power of ensemble could be a real number, we take for example $\mathbb{R^{\frac{1}{2}}}$, Does this meant the space dimension is $\frac{1}{2} $? Note:I want $\mathbb{R^{\frac{1}{...

linear-algebra vector-spaces metric-spaces dimension-theory
 
 
1 hour later…
Martin Sleziak
Martin Sleziak
8:25 AM
The tag-excerpt and the tag-info for shuflles and also the tag-excerpt and tag-info for chain-complexes have been created by Javi.
@MarkMcClure: If everyone casting a vote also left a comment as yours above (and below the other answer), do you realize how long this webpages would become? Also, I see much more votes on the question that on the answers, which is strange. The OP should leave it open for longer than usual. — Alex M. 2 hours ago
@AlexM. On the other hand, Mark McClure was specifically asked to look at this discussion - as one of the local experts in topics related to fractals - there is certainly some overlap between the questions currently tagged (dimension-theory) and fractals. (And knowing the position of the top users from this tag on the issue might be useful.) — Martin Sleziak 2 hours ago
@AlexM. Martin is correct - I was simply communicating directly with those users who had communicated with me in chat. Leaving superfluous comments is not my usual behavior, which is why I have more than twice as many votes as comments on Meta. I suppose, though, that the comments are no longer necessary so I've removed them. I invite you to do the same. — Mark McClure 2 mins ago
Just for the record, @MarkMcClure's comment (now deleted) was along the lines that he supports Xander Henderson's suggestions for having a separate tag for dimension theory in analysis and dimension theory in algebra. (I do not remember the exact wording of the comment.)
@Andrews Yeah, but these notions of dimension are not simply metric; they also typically involve measures, and are properly contained in the the field of analysis. "Analytic dimension theory" seems like the right terminology to contrast with "algebraic number theory." The adjective "topological" doesn't seem quote right to me, though if it came down to it, I suppose that "topological" might be an acceptable alternative adjective. — Xander Henderson 2 days ago
@XanderHenderson I guess this was a typo and you mean algebraic dimension theory rather than algebraic number theory, right? Or did you want to suggest some analogy between division of dimension-theory tag and the fields analytic number theory and algebraic number theory?
 
Martin Sleziak
Martin Sleziak
9:15 AM
The thing that worries me about this proposal the most is that it suggests to put under the same tag questions about dimension of vector spaces and dimension of rings. I am not sure whether a tag for dimension in linear algebra is needed, but if it is, I think it should be separate from the tag intended mostly for rings and modules. — Martin Sleziak 10 secs ago
Apr 18 at 9:48, by YuiTo Cheng
Regardless of whether the division of the tag dimension-theory is needed, it really has nothing to do with linear-algebra or vector-spaces.
Apr 19 at 8:42, by Arnaud D.
This makes me think that we need to have a tag for dimension in the sense of linear algebra, if only because users will want to tag their questions "dimension". Would this fall under the suggested "dimension-theory-algebra" tag?
Krull dimension for commutative rings is the supremum of lengths of chains of prime ideals (by inclusion). It is unrelated to the dimension of a vector space. — hardmath 2 days ago
 
 
1 hour later…
Martin Sleziak
Martin Sleziak
10:32 AM
@XanderHenderson It seems to me a bit unclear from your answer whether or not you want to include dimension of vector spaces in the new tag.
On one hand you mentioned:
> Finally, there are a lot of very elementary questions which show up in the dimension-theory tag which really don't belong. These tend to be of the type "What is the dimension of this subspace of a vector space?"
But in the same post you say that (emphasis mine):
> algebraic-dimension-theory, for questions about algebraic notions of dimension (e.g. dimension of a vector space, Krull dimension).
Would you mind clarifying this a bit?
As I've mentioned, my impression is that the notion of dimension in linear algebra/vector spaces should be kept separate.
2
 
 
2 hours later…
Xander Henderson
Xander Henderson
12:45 PM
@MartinSleziak Oi, vey. Yup.
@MartinSleziak Elementary questions (e.g. those from a basic linear algebra class) certainly do not belong. However, there are, I believe, more advanced questions about abstract vector spaces which might, potentially, be relevant. That being said, I am not an algebraist.
My goal is to separate out the stuff which I feel belongs (e.g. dimensions in metric spaces, plus related notions of topological dimension) from everything else. People who are more expert in those other things should probably figure out the boundaries of the new tag or tags.
 
 
7 hours later…
Martin Sleziak
Martin Sleziak
7:36 PM
@XanderHenderson I have posted a new answer where I tried to explain my objections to having dimension of vector spaces in the same tag. We'll see what others have to say about this.
2
I agree with you that it would be nice to hear form somebody who is expert in abstract algebra, ring theory, commutative algebra. Unfortunately, tag-related discussions usually do not attract attention of many users.
 
 
2 hours later…
Xander Henderson
Xander Henderson
9:36 PM
@MartinSleziak Thank you for posting a new answer. In light of the comments provided below my own, I think that the proposal needs some work, but I am hesitant to edit the answer for fear of invalidating votes which have been cast.
 

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