The Banach-Steinhaus theorem (Uniform Boundedness Principle) states that for a family of bounded linear operators T_α _(α ∈ A) from a Banach space X to a normed space Y, pointwise boundedness implies uniform boundedness. Specifically, if \sup_(α ∈ A) \|T_α(x)\| < ∞ ∀ x ∈ X then \sup_(α ∈ A) \|T_α\|_(L(X,Y)) < ∞. Does this result still hold if X is only assumed to be a normed space rather than a Banach space? If not, what is a counterexample and which specific failure of completeness allows the theorem to break?