Application of a result on some bounded functionals on a subspace of $C([0,1])$

Question

The following result was proved in a previous post: Bounded functionals on Banach spaces.

Let $(X, \|.\|)$ be a Banach space such that

• $X \subset C([0,1]) $
• For every $r\in \mathbb{Q}\cap[0,1], f\mapsto f(r)$ defines a bounded linear functional on $X$.

There exists a $C>0$ such that, for all $f\in X$,
$$\sup_{x\in[0,1]} |f(x)| \leq C\|f\|.$$

Question:

Does anyone know an example of space $X$ where this result is interesting?

Indeed I feel that the example above could be a very nice application of the Banach-Steinhaus theorem, but the examples of spaces $X$ I thought of were too simple:

1. One could easily prove the result without the Banach-Steinhaus theorem.
2. The assumption on boundedness would be easy to prove for all $r$ in $[0,1]$.

If someone has an example of space $X$ satisfing the first point, even without the second one, I am already interested.