Understanding notation of a function space from Nestorov's Convex Optimization textbook

I am looking at Theorem 2.1.5 in the book Lectures on Convex Optimization, by Yuri Nestorov, where the theorem statement is for all $f\in \mathcal{F}_L^{1,1}(\mathbb{R}^n, \|\cdot\|)$ and I don't understand the function space $\mathcal{F}_L^{1,1}(\mathbb{R}^n,\|\cdot\|)$.

My attempt at understanding:Above equation 2.1.7, the book says: we introduce a new notation:$f \in \mathcal{F}^{1,1}_L (Q, \cdot )$means that$Q \subseteq \text{dom}\,f$and$\| \nabla f(x) − \nabla f(y) \|_\star ≤ L\|x − y\|$ , $\forall x,y \in Q$.

The dual norm is defined above equation 2.1.6 as$$\|g\|_\star = \max_{x\in \mathbb{R}^n} \{ \langle g,x \rangle: \|x\|\leq 1 \}$$

So I don't understand whether in the function space $\mathcal{F}^{1,1}_L (\mathbb{R}^n, \|\cdot\| )$, $\|\cdot\|$ refers to Euclidean norm or some other norm and if in the definition of the dual norm, would the inner product correspond to the standard dot product of vectors in $\mathbb{R}^n$.

It's been a long time since I have looked at convex functions, so please bear with my basic question. Thanks.