A question from definition of coercive function.

Question

I have one doubt about the definition of Coercive function:

Definition: if for a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$:
$$ \lim_{\|x\| \to +\infty} f(x) = +\infty, $$ then $f$ is a coercive function.

As I checked in the literature, it is defined on $\mathbb R^n$; is it true it only defines on $\mathbb R^n$ or we can define also on its subsets? Now, could we define a coercive function on a domain like $x \in [0, +\infty)\subset \mathbb{R}$? For example, please clearly determine below each one is coercive or not:

1. $f(x) = x^2$ for $\text{dom}(f) = \mathbb{R}$; coercive or not coercive?
2. $f(x) = x^2$ for $\text{dom}(f) = [0,+\infty)$; coercive or not coercive?
3. $f(x) = x^2$ for $\text{dom}(f) = [0,1]$; coercive or not coercive?

Thanks.

Answer 1

The definition is meaningful in any topological space. In normed spaces, it requires the possibility for the domain to include elements of arbitrarily large norm. Hence the definition can be formally given in any subset that... "contains infinity". It can be written as $$ \lim_{\substack{\|x\| \to +\infty \\ x \in X}} f(x) = +\infty, $$ where $X$ is the domain of $f$.