A function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is called coercive if $$ \frac{f(x)\cdot x}{\|x\|} \rightarrow \infty \;\; \text{as} \;\;\|x\|\rightarrow \infty.$$
I came across this requirement in calculus of variations, where a coercivity condition is needed to show that a sequence of functions gamma-converges to some limiting function. It seems related to showing that the compactness condition that usually accompanies Gamma-convergence - does coerciveness imply compactness? I don't see how they are related, but they seem to be.
Edit: The "compactness condition" I refer to is that any sequence of arguments $\{u_n\}_{n\in \mathbb{N}}$ has a convergent subsequence. I think that this compactness condition is implies equi-coerciveness of the sequence $\{u_n\}$. I don't know why coerciveness is used in proofs of gamma-convergence instead of compactness directly. Anyways, I still want to know why the functions are called "coercive".