Operator in fractional spaces

Question

I am no mathematician so please excuse me if I don't use the right terminology. Lets say I have a function $y=f(x): x \in \mathbb{R}^3$ and $y \in \mathbb{R}^1$. So this function "projects" points from a higher dimensional space into a lower dimensional space.

I was wondering wether you can somehow divide this projection in smaller steps, i.e, interpret this function application as an equivalent composition of several "shorter projections":

$\mathbb{R}^3 \Rightarrow \mathbb{R}^2 \Rightarrow \mathbb{R}^1$ or in general:

$\mathbb{R}^m \Rightarrow \mathbb{R}^n = \mathbb{R}^{k_3} \circ \mathbb{R}^{k_2} \circ \mathbb{R}^{k_1} \circ ...$ where $\sum \Delta k_i = m-n$.

Taking this to the continuum case may be there is an operator $G$ and an integral like operation(for function composition) such that:

$\int^n_m G(k)dk = F(n) - F(m)$

This operator $G$ could be interpreted as an infinitesimal projection transformation from dimension $k$ to dimension $k+dk$.

I have found that fractional spaces have something to do with this but I haven't seen anything similar to the G operator I am referring to here. Still I hope this has already been studied

Question: What is the name of this branch of math? I would be very thankful if you could direct me to some resources to see what the main results in this field are.

Edit: I know there are multiple ways to go from higher dimensional spaces into lower ones. I just want to impose conditions on the way this is done, (sort of a variational calculus where the functionals infinitesimally project between dimensions and we are trying to find some "optimum way"(optimality criteria still to be defined) of going from $\mathbb{R}^m \Rightarrow \mathbb{R}^n$) and I was looking for a theoretical foundation to start with.

Answer 1

Your ideas remind me a notion of inverse limit (unfortunately, I find no simpler description of it).

Answer 2

I fear that your question is not well enough formulated to quite make sense.

First, if you have a function $f$ from, say, $\mathbb R^4 $ to $\mathbb R^1$, it can always be decomposed into a sequence of maps into successively smaller spaces. For instance, if \begin{align} f(x, y, z, w) &= x^2y + z \sin w \end{align} then I can define \begin{align} f_3 &: \mathbb R^4 \to \mathbb R^3 : (x, y, z, w) \mapsto (x^2 y + z \sin w, 0, 0) \\ f_2 &: \mathbb R^3 \to \mathbb R^2 : (a, b, c) \mapsto (a, b) \\ f_1 &: \mathbb R^2 \to \mathbb R^1 : (p, q) \mapsto p \end{align} and we'll have $$ f = f_1 \circ f_2 \circ f_3. $$ Note, too, that the dimensions of the intermediate spaces are 3, 2, 1, so their sum is 6, not 4-1 = 3. So the formula just before "taking this to the continuum case" doesn't quite work.

That means that your continuum analog also doesn't really quite make sense.

(By the way, there's nothing special about the dimensions $4$ and $1$ here: if you have a map from from $\mathbb R^n$ to $\mathbb R^k$, with $k < n$, it can always be factored into a sequence of $n - k$ maps analogous to the ones I wrote above.)

I'd like to think that you're doing something interesting here, but so far, I just can't see it. As a starting point, I'm going to recommend that you (a) learn a widely accepted notation for writing down functions, so that you can communicate clearly with mathematicians, and (b) learn enough linear algebra to have a firm grasp on the concept of "dimension" in at least that context, so that you can avoid simple mistakes like the one just above your "Taking this to the continuum case" sentence, and can think up answers like the one I presented here to your question about factoring dimension-reducing functions through one-dimension-at-a-time projections.