The aim of this page is to list examples of bounded and unbounded linear operators.

Bounded

• Identity operator, Zero operator
• Shift operators on $\ell^p$
• A linear operator is continuous if and only if it is bounded (see this page).
• Any isometry is bounded.
• A multiplication operator $h(t) \mapsto f(t) h(t)$ , where $f(t)$ is continuous and $h\in L^p[0,1]$ .
• An integral operator $h(t) \mapsto \int_0^1 K(t,s) h(s)\,ds$ , where $\int_0^1\int_0^1 \abs{K(s,t)}^2\,ds\,dt < \infty$ and $h\in L^2[0,1]$ . In fact this is a Hilbert-Schmidt operator.
• The Volterra operator $h(t) \mapsto \int_0^t h(s)\,ds$ , where $h\in L^p[0,1]$ .

Unbounded

• The derivative is an unbounded operator on the vector space of smooth functions equipped with the $\operatorname{sup}$ -norm.