Here is a sufficient condition for a linear operator to be bounded:

Theorem (Hellinger-Toeplitz theorem)

Let be an everywhere defined linear operator on a Hilbert space with for all and in . Then is bounded.

This theorem states that an everywhere defined linear operator on a Hilbert space that is symmetric everywhere on is always bounded. The Hellinger-Toeplitz theorem implies that an unbounded symmetric operator cannot be defined on all of ! It tells you that when dealing with unbounded operators it is very important to specify the domain on which the operator is defined.