The problem is formulated using old terminology and I want to understand what it actually says.

The problem reads: "A space $E$ of type (B) has the property (a) if the weak closure of an arbitrary set of linear functionals is weakly closed. [A sequence of linear functionals $f_n(x)$ converges weakly to $f(x)$ if $f_n(x) \to f(x)$ for every $x$.] The space $E$ of type (B) has the property (b) if every sequence of linear functionals weakly convergent converges weakly as a sequence of elements in the conjugate space $\bar{E}$. Question: Does every separable space of type (B) which has property (a) also possess property (b)?"

I know that "space of type (B)" means "Banach space" in modern terminology.

Q1: Does "linear functional" include the assumption of continuity?

Q2: What is the conjugate space? Does it mean the dual space of continuous linear functionals on $E$?

If the answers to Q1 and Q2 are yes, then this would mean Banach spaces with property (b) are exactly what nowadays are called Grothendieck spaces. [A Grothendieck space is a Banach space $E$ such that every sequence in the dual space $E^*$ that converges weak* - that is, with respect to $\sigma(E^*,E)$ - also converges weakly - that is, with respect to $\sigma(E^*,E^{**})$.]

I am also unsure how to understand the condition "the weak closure of an arbitrary set of linear functionals is weakly closed", as this seems like a tautology. Is the point here that we consider a sequential closure?