Uniform Boundedness Principle

A ``pointwise-bounded'' family of continuous linear Operators from a Banach Space to a Normed Space is ``uniformly bounded.'' Symbolically, if $\sup \vert\vert T_i(x)\vert\vert$ is Finite for each $x$ in the unit Ball, then $\sup \vert\vert T_i\vert\vert$ 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.