I read this definition in a book about variational inequalities of Kinderlehrer and Stampacchia.
The concept of mapping continuous on finite dimensional subspaces is introduced in Definition III.1.2.
Definition 1.2. The mapping $A$ from $\mathbb K\to X'$ is continuous on finite dimensional subspaces if for any finite dimensional subspace $M$ a $X$ the restriction of $A$ to $\mathbb K\cap M$ is weakly continuous, namely, if $$A \colon \mathbb K\cap M \to X'$$ is weakly continuous.
The concept of weak continuity is mentioned in the definition, but I don't think it is related to the usual weak topology defined here.
Thank you very much for any hints.
The authors mean the weak* topology on $X'$, i.e., the topology induced by the family of seminorms $p_x(x^*) = |\langle x^*, x\rangle|$, $x\in X$. So, for every finite dimensional space $M$ the map $K\cap M\to X'$ must be continuous with respect to the standard topology on $M$, restricted to $K\cap M$, and the weak* topology on $X'$.
Another way to express the above: for every $x\in X$, the scalar function $z\mapsto \langle Az, x\rangle$ is continuous on $K\cap M$. You can see this fact being used on page 85, soon after the definition.