Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$, and $\|\cdot\|_*$ be its dual norm. For any nonzero vectors $u,v \in \mathbb{R}^n$, define \begin{equation} A(u,v) = \arccos \frac{u^T v}{\|u\|\|v\|_*}. \end{equation} Question: Does $A(u,v)$ have a name? Any reference will be highly appreciated.
Extended question: Let $(X, \|\cdot\|)$ be a Banach space, and $(X^*, \|\cdot\|_*)$ be its dual space. For any nonzero vectors $x\in X$ and $x^* \in X^*$, define \begin{equation} A(x,x^*) = \arccos \frac{x^*(x)}{\|x\|\|x^*\|_*}. \end{equation} Does $A(x,x^*)$ have a name?
Remark: The original question is indeed a special case of the extended one. Consider the Banach space $(\mathbb{R}^n, \|\cdot\|)$. Let $((\mathbb{R}^n)^*, \|\cdot\|_D)$ denote its dual. For each $x^* \in (\mathbb{R}^n)^*$, there exists a unique vector in $\mathbb{R}^n$, denoted by $\tau(x^*)$, such that \begin{equation} x^*(x) = x^T \tau(x^*). \end{equation} $\tau$ is a linear bijection between $(\mathbb{R}^n)^*$ and $\mathbb{R}^n$. In addition, $\tau: ((\mathbb{R}^n)^*, \|\cdot\|_D) \rightarrow (\mathbb{R}^n, \|\cdot\|_*)$ is an isometry, because \begin{equation} \|x^*\|_D = \sup_{0\neq x\in\mathbb{R}^n} \frac{x^*(x)}{\|x\|} =\sup_{0\neq x\in\mathbb{R}^n} \frac{x^T \tau(x^*)}{\|x\|} =\|\tau(x^*)\|_*. \end{equation} Thus $\tau$ is an isometric isomorphism between $((\mathbb{R}^n)^*, \|\cdot\|_D)$ and $(\mathbb{R}^n, \|\cdot\|_*)$. Under this isomorphism, the original question can be regarded as a special case of the extended one.
Background (well, this information is not useful for answering the question, but I do not want this question to be marked as "off-topic" for lack of background information): I am a computational mathematician writing a paper on the convergence analysis of a numerical scheme. The quantity $A(\cdot,\cdot)$ arises naturally in my analysis. $A(\cdot,\cdot)$ is obviously a generalization of the angle in $\mathbb{R}^n$, but it would be nice if this quantity has a commonly accepted name.
Thank you very much!