A ``pointwise-bounded'' family of continuous linear Operators from a Banach Space to a Normed Space is ``uniformly bounded.'' Symbolically, if is Finite for each in the unit Ball, then is Finite. The theorem is also called the Banach-Steinhaus Theorem.

**References**

Zeidler, E. *Applied Functional Analysis: Applications to Mathematical Physics.*
New York: Springer-Verlag, 1995.

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1999-05-26