Recently, I have seen the so-called uniform boundedness theorem, which says:

> Let $(X,∥⋅∥)$
 be a Banach space and $(Y,∥⋅∥)$
 be a normed linear space. Let $A⊂B(X,Y)$
 be a pointwise bounded family of bounded linear transformations from $X$
 to $Y$
. Then the family $A$
 is uniformly bounded.

I was wondering if there is any 'simple' non-Banach space that verifies this property: that is, that **for all** choices of $Y$, a pointwise bounded family of bounded linear transformations is uniformly bounded.

I have thought that maybe some kind of sequence space might work, but I have not quite found one that does.