What Are The Formal Definitions Of A Dimensionality Space And The Space Spanning Those Spaces?

Question

[Egon Willighagen]

I am looking for a formal definition of the concept dimensionality space. I know what it is, but looking for a solid, citable definition. You get bonus points if you can point me to the correct term for the space defining the various dimensionality spaces. Seriously!

To clarify things, I am not asking what dimensionality is. I am also not asking what the concepts mean, but a citable definition.

[Jarretinha]

Just to clarify: Are you asking for the dimensionality as in topology/algebra or that one in data mining/language processing?

[Jeroen Van Goey]

You will maybe have more success if you ask this question at http://mathoverflow.net/

[Egon Willighagen]

"Dimensionality space" as from vector algebra.

Answer 1

[User 0339]

I doubt there's such a thing. You can say a reduced-dimensionality space, perhaps, but in that case "dimensionality" is attached to "reduced" not to "space".

"Reduced-dimensionality" space is a space which has a smaller dimension with respect to some other space.

What's the dimension of a space?

Typically, one refers to the dimension of a vector space.

1 - A vector space is a set of elements called vectors such that the sum of two elements is still in the set, and one can multiply each element by a real number and still get an element of the set.

2 - A subset (of cardinality k, with elements v_1,...,v_k) of a vector space V is said to be linearly independent if there are no nonzero real numbers c_1,...,c_k such that c_1*v_1+...+c_k*v_k=0

(remember that you can multiply vectors by numbers, and you can sum vectors, so the above makes sense. 0 means the 0 vector.)

3 - The cardinality of the biggest possible such subset of V is the dimension of V.

Example:

For example you can give the plane the structure of a vector space by choosing on it coordinates (x,y).

one possible linearly independent subset of the plane is given by the two vectors (1,0) and (0,1). Indeed it is not possible to find non-zero c_1,c_2 such that

c_1*(1,0)+c2*(0,1)=(0,0)

It turns out that you cannot build bigger linearly independent subsets of the plane : hence it is a vector space of dimension 2.

You can read this: http://en.wikipedia.org/wiki/Linear_independence and go on hopping on Wikipedia.

Answer 2

[Jarretinha]

This set would not be a space at all. What would be your metric? Or your norm? What we normally do in bioinformatics is to work with pre-topologies. They represent some sort of "sequence space" closed under edition/recombination. In a practical sense, reversible-jump Markov chains deal with variable dimensionality of parameter space and there is some theory attachted to them. Nevertheless, if you want more information about spaces and similar constructions I do recommend you to search on http://lib.org.by for the book of Kolmogorov on functional analysis or similar treatises on abstract spaces.

Answer 3

[Michael]

Any basic linear algebra textbook should do fine for a citation. Btw, what is the connection to bioinformatics?

You are asking for the definition of dimensionality of vector-spaces? The (number of vectors that form) cardinality of a basis for the vector space. For example in R^3 each basis consists of three vectors, eg. the standard basis (0,0,1), ... http://en.wikipedia.org/wiki/Dimension_%28linear_algebra%29

Edit: Now I think I know what you mean, but still, the isolated use of dimensonality space is incorrect both in the gramatical and algebraic sense. Try a google search for "dimensionality space" inlcuing the quotes. Dimensionality is a feature of a vector-space and there is no dimensionality space (such as a "space of dimensions").

Reduced dimensionality space seems to be used eg in the sense of dimension reduction, such as "Our great method reduces the 42-input space to a problem in a reduced dimensionality space (eg of dimensionality 3)"

Look for example at: e.g. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.91.4068

That might be an example headline that you might be thinking of.

[Egon Willighagen]

What you are saying is that dimensionality space == dimensional space ? I'm not asking what the dimensionality of a space is.

[Egon Willighagen]

The connection to bioinformatics is, that moving between dimensionality spaces is what you do all the time when doing bioinformatics.

[Michael]

Sorry, then I don't understand what this is about. I have never heard of a "dimensionality space", and I have never moved between them, I'm quite sure ;o) sorry :

[Egon Willighagen]

No worries... :) What about the second question... let's assume a 'dimensionality space' is identical to a 'n-dimensional space' (I think it is, and I asked for a formal definition to confirm this use in literature), what would you call the space that spans these n-dimensional spaces? There is the Hilbert space for functions with certain properties... what would a space be called containing 0-dimensional space, 1-dimensional space, 2-dimensional space, ... , n-dimensional space?

[Michael]

That would be a n-dimensional space ;)