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.

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.

EDIT: I found in a book that the space consisting on the real sequences that take on finitely many distinct values is an example, but it does not include a proof. Is this proven easily?