In this paper, page $131$, in the proof of Proposition $4.1$, there is this sentence:
We first note that if $V$ is a separable Banach space, the subset of Lip$(V)$ consisting of all weakly continuous functions is norming.
Question: What is the definition of norming?
From the book 'Topic in Banach Space Theory', Chapter $1$, page $17$, the author mention norming in the following sentence (Remark $1.4.2$ (b)):
Let us observe as well that if $X$ is separable then not only is the sequence $(x^*_n)_{n \in \mathbb{N}}$ in $(ii)$ separating for $X$ but it is also norming in $X$. That is, the norm of any $x \in X$ is completely determined by this numerable set of functionals:
$$\| x \| = \sup_{n}{|x^*_n(x)|}, x \in X$$
If the norming in the paper has the same meaning as in the book, then how do we interpret its meaning?
How to show the subset is norming?