The uniform boundedness principle says if we have a collection of bounded linear operators $\Gamma$ from a banach space $X$ into a normed vector space $Y$, which is pointwise bounded on $X$, i.e. $$\sup \{\|Tx\| : T \in \Gamma \} < \infty \, \forall x \in X,$$ then $\Gamma$ must be uniformly bounded, $$\sup \{\|T\| : T \in \Gamma \} < \infty.$$
Now, I wonder is there some kind of converse statement? So, if $X$ is a normed space and the following holds for every normed space $Y$:
$$\Gamma \subset \mathcal{L}(X,Y) \text{ pointwise bounded on }X \Rightarrow \Gamma \text{ uniformly bounded},$$
where $\mathcal{L}(X,Y)$ is the space of bounded linear operators from $X$ to $Y$.
Then does it follow that $X$ is a banach space?
Edit: Or are there any partial converses?
Edit 2: I think I need to clarify further. The converses I am looking are not of the form "uniformly bounded $\Rightarrow$ pointwise bounded".
Instead we should have a space $X$, more general than a Banach Space, in which the uniform boundedness principle holds. So every pointwise bounded collection $\Gamma$ of linear bounded operators (into an arbitrary space $Y$) must be uniformly bounded. And this should then imply that $X$ is a Banach space. It's okay if the theorem requires more conditions, than just the uniform boundedness principle, on $X$ for the conclusion to hold.
Best Answer
Note: Here are two examples of partial converses which could be convenient.
We can read in section $27$: Uniform boundedness
And later on in section $51$: Uniform boundedness of linear transformations
We can read in section $18.2$: The weak Topology:
Maybe also interesting is the following generalisation to Hausdorff locally convex topological vector spaces stated in Appendix C: The Uniform Boundedness Principle
Note: The conclusion of uniform boundedness on bounded sets in the classical uniform boundedness principle for normed spaces is equivalent to the condition that the family of operators is equicontinuous with respect to the original topology of the domain spaces. See e.g. this paper by Ronglu Li and Charles Swartz.