# Notation for the set of all real spaces of abitrary size

What is the general notation for a function $$f$$ that maps a point in $$n \in \mathbb{N}$$ to a set in $$\mathbb{R}^n$$.

$$f : \mathbb{N} \mapsto ?$$

What is ?

• I think that your question it's no clear. You can write how do you like it, for example $f: \mathbb{N} \to \mathbb{R}^{n}$. – Александр Пальма Oct 23 at 13:40
• yes but the output dimension depends on the input so n=1 gets mapped to R^1 and n=2 gets mapped to R^2. – user3680510 Oct 23 at 13:40
• So... you have an input, I'll call $a$ which is a natural number and you want $a$ to output to a set in $\Bbb R^b$ where $b$ is necessarily equal to $a$? Letting $Y = \mathcal{P}(\Bbb R^1)\cup \mathcal{P}(\Bbb R^2)\cup \mathcal{P}(\Bbb R^3)\cup\dots$ perhaps you mean to have $f~:~\Bbb N\to Y$. Note that all we are interested in this part of the notation is the codomain. We aren't interested in the fact that $f(a)$ must specifically map to a set in $\Bbb R^a$. That aspect of it can be handled later when we define more specifically $a\mapsto ?$ – JMoravitz Oct 23 at 13:50
• @JMoravitz yes $a=b$ – user3680510 Oct 23 at 13:51

You can go with $$f:\Bbb N\to \mathcal P\left(\bigcup_{n\in\Bbb N}\Bbb R^n\right)$$ and then specify the additional condition that $$f(n)\subseteq\Bbb R^n$$ for all $$n$$, or you can be arguably a little more obscure and say that $$f\in\prod\limits_{n\in\Bbb N}\mathcal P(\Bbb R^n)$$. Personally, I don't know which one to recommend, but probably in general I'd choose the first one.
• and what is the reason i need the $\mathcal{P}$? – user3680510 Oct 23 at 14:01
• You said "a function $f$ that maps a point in $n \in \mathbb{N}$ to a set in $\mathbb{R}^n$", so the codomain must be some kind of power set. – Gae. S. Oct 23 at 14:07