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.