Every linear operator between finite-dimensional normed spaces is compact because it is automatically continuous (equivalently bounded), which makes the image of the unit ball bounded. Taking the closure of this bounded image yields a closed and bounded set, and in finite-dimensional spaces, closed and bounded sets are compact.
Conditions: The operator is linear.; The domain and codomain are finite-dimensional normed spaces.
Every linear operator between finite-dimensional normed spaces is compact because it is automatically continuous (equivalently bounded), which makes the image of the unit ball bounded. Taking the closure of this bounded image yields a closed and bounded set, and in finite-dimensional spaces, closed and bounded sets are compact.
Conditions: The operator is linear.; The domain and codomain are finite-dimensional normed spaces.