Let $X$ be a normed vector space over $\mathbb R$, not necessarily Banach. Let $X'$ denote the dual of $X$, that is, the set of all bounded, linear functionals on $X$: $$X'\equiv\{f:X\to\mathbb R\,|\,\text{$f$ is linear and bounded}\}.$$ Suppose that $A'\subseteq X'$ is a topologically bounded set in the weak* topology on $X'$. This means that for any $x\in X$, there exists some $M_x>0$ such that $|f(x)|\leq M_x$ for all $f\in A'$.
Question: Is $A'$ necessarily bounded in the norm? That is, does there exist some $M>0$ such that $\|f\|\leq M$ for any $f\in A'$ (where $\|\cdot\|$ is the operator norm on $X'$)?
The answer is affirmative if $X$ is a Banach space, as can be shown easily using the uniform boundedness principle. Is this also true is $X$ is not a Banach space? If not, do there exist some easy counterexamples?