Exercise:
Let $X$, $Y$ be normed spaces and $A:X\to Y$ be a linear and continuous operator, which has the property \begin{equation} \exists C>0 \text{ s.t. } \forall x\in X \quad \Vert Ax\Vert\geq C \Vert x \Vert \end{equation} Can A be a compact operator?
I guess, yes it can. But how to show it? I'm confused because $A$ is not bounded, and intuitively I would to use Arzela-Ascoli.
In fact, $A$ can't be a compact operator (if $X$ is infinite dimensional). This condition is sometimes referred to as "$A$ is bounded away from $0$".
To show that it can't be compact, it suffices to show that there is a bounded sequence whose image has no convergent subsequence. Note (by the usual application of Riesz's lemma) that there exists a bounded sequence in $X$ such that the distance between any two elements in the sequence has a lower bound.
On the other hand, if $X$ is finite dimensional, then every operator on $X$ is compact. If $X$ is infinite dimensional and $Y$ isn't, then the operator can't have this property.