In the uniform boundedness principle(the Banach-Steinhaus theorem), the domain is most generally given as an F-space(i.e. a complete quasinormed topological vector space). Here, pointwise boundedness is equivalent to uniform boundedness, which is in turn equivalent to equicontinuity.
But, for general normed linear spaces $X$ and $Y$, and for $F$ which is a collection of continuous linear maps from $X$ to $Y$, how can it be shown that uniform boundedness is equivalent to equicontinuity? I can just accept the fact intuitively and visually, but cannot move my hands to prove it mathematically with rigor. Any comment would be welcome.