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I would like to give counterexamples to show that the uniform boundedness principle fails if one relaxes the assumptions in any of the following ways:

  1. The given space is merely a normed vector space rather than a Banach space (i.e. completeness is dropped).

  2. The family of linear operators are not assumed to be continuous.

  3. The family of continuous operators are allowed to be nonlinear rather than linear.

Thank you for all the comments.

1 Answers 1

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1) Let $X$ be the linear span of $e_1, e_2, \ldots$, with your favourite norm such that $\|e_j\| = 1$, and $T_n: X \to X$ with $T_n e_j = j e_j$ for $j \le n$, $0$ otherwise.

2) Take one discontinuous linear operator on a Banach space.

3) You have to be careful with precisely how you state the UBP to have it make sense at all for nonlinear operators.

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    Dear Sir. I meet difficuty in a question in the following link (http://math.stackexchange.com/questions/474702/vector-of-reduction) Could you help me to give your idea and comments?2013-08-27