Examples:

- a 2x2 matrix can represent a linear map from $R_{2}$ to $R_{2}$, so which is a linear operator
- the derivative is a linear map from $C_{∞}$ to $C_{∞}$, so which is also a linear operator

Given a linear operator $A$ over a space $S$ that has a inner product defined, we define the adjoint operator $A_{†}$ (the $†$ symbol is called "dagger") as the unique operator that satisfies:

$∀v,w∈S,<Av,w>=<v,A_{†}w>$