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Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.

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You have a collection $(T_i)_{i\in I}$ of bounded operators, and $(e_j)_j$ a collection of directions. Given a direction, the "orbit" $\{T_ie_j,i\in I\}$ is bounded. This means that there is a bounded set $S_j$ such that applications of the bounded operators to $e_j$ cannot make you leave this set. If the space is "without holes", this was because the operators "had a bounded amplitude".

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Could you please expand on the last part of this answer? – Tyler Hilton Dec 6 '13 at 20:17

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