DE
Updated May 15, 2026
Question details
Does the Banach-Steinhaus theorem hold for non-complete normed spaces?
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?
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