Characterize the continuity of linear maps between Banach spaces in terms of continuous linear functionals

Question

I solved a) and b). But I cant seem to get a grip of what characterizes the functionals for which we want continuity in the general case. Hints please!

Answer 1

The families of functionals mentioned in (a) and (b) have a property in common: they separate the points of the Banach space, meaning that for every nonzero element of the space the family contains a functional that does not vanish on that element.

In general, suppose $\mathcal F\subset Y^*$ is a family that separates the points of $Y$. Then a linear map $M:X\to Y$ is continuous if and only if $f\circ M$ is continuous for every $f\in \mathcal F$.

That was the hint. The proof is under spoilers.

One direction is trivial. To prove the other, use the Closed Graph Theorem.

Suppose $x_n\to x$ and $Mx_n\to y$; the goal is to show $Mx=y$. On one hand, $f(Mx_n)\to f(Mx)$ because $f\circ M$ is continuous; on the other, $f(Mx_n)\to f(y)$ because $Mx_n\to y$. So, $f(Mx-y)=0$ for all $f\in\mathcal F$, which implies $Mx-y=0$.