Prove by giving an example that completeness is significant in Banach-Steinhaus theorem. I need to give an example of space and family of operators to prove that completeness of space $X$ in Banach-Steinhaus (the uniform boundedness principle) is in fact needed.
I guess space $c_{00}$ would be one of proper places to look for the needed operators, but it's my intuition.

Yes, $c_{00}$ (the space of sequences with finitely many nonzero terms) is the prime candidate for counterexamples involving lack of completeness. The family of linear functionals $f_n(x) = nx_n$, $n\in\mathbb{N}$, provides such a counterexample: bounded pointwise but not bounded in the norm.
