ABSTRACT
A Principle of Uniform Boundedness for Linear Functionals: If (fn) is a sequence of continuous linear functionals on a Banach space X and for each x ∈ X there exists Mx such that |fn (x)| ≤ Mx for every n then the sequence (‖fn‖) of norms is bounded. As the latter bound does not depend on any particular point, it is a “uniform” bound. Alternatively, if {fn} is σ (X ′, X)-bounded, then it is norm bounded or: the concepts of norm and weak-∗ boundedness coincide.
