The term "unbounded set" is usually defined as such: if a set is not bounded, then it is unbounded.
This definition, while having its own merits, is somewhat weak. For example, in $\mathbb R^2$, the set $\{(x,y)|1<x<2\}$ is an unbounded set, even though it is actually bounded by two lines: $x=1$ and $x=2$.
"Bounded by a hyperplane" is defined in the fashion of convex analysis: a set S is bounded by a hyperplane $\langle \cdot,v\rangle=c$ if all points in S are at the same side of the hyperplane:
$\langle x,v\rangle\leq c $ $\forall x\in S$ or $\langle x,v\rangle\geq c$ $\forall x\in S$.
Now we are going to have a stronger definition called "strongly unbounded":
A set S is strongly unbounded if S is not bounded by two non-intersecting lines at the different side.
Does this definition make sense to you? Anyone defined this object before? Are there any interesting properties?
Precise definitions added! Thank you for all your comments. Any answers are welcome; no need to be complete.