What is the meaning of $\|\cdot\|$

Question

I am reading a paper, and I saw the following symbol

$\|\cdot\|$ in it. I understand that the vertical bars mean norm. But the dot inside is is confusing me.

The rest of the paragraph reads

We denote $\|\cdot\|_Q$ by $l_q$. However, be careful: $l_q$ is not a norm. Subadditivity (triangular inequality) is the only one of the three properties required by a norm, which is satisfied by $lq$.

Answer 1

Usually every mathematician writes $|| \cdot ||$ instead of $||\;||$. The norm $|| \cdot ||$ is actually a function. One always write $||x||$ instead of $||\cdot||(x)$

So the dot in the middle denote just the argument as mentioned in the comment!

Answer 2

The usual convention in mathematics is to name functions by single letters or short abbreviations, then put the argument of the function in parentheses to the right of the function's name. For example, if $f$ is a function defined on some domain, then the value of $f$ at some point $x$ is written $f(x)$.

However, there are times when this notation is inconvenient. For example, suppose that $f : \mathbb{R}^2 \to \mathbb{R}$. The usual convention would be to write $f(x,y)$ to denote the value of $f$ at the point $(x,y)$. But sometimes we want to consider $f$ as a function of just a single variable—for example, when computing the partial derivative $f_x$, we think of $y$ as a fixed constant. If we want to emphasize the fact that $f$ doesn't really depend on $y$, but retain our usual notation, then we might write $$ f(\cdot,y) : \mathbb{R} \to \mathbb{R}.$$ The dot indicates the place where the variable is "plugged in." Similarly, if we have a normed space $(X,\|\cdot\|)$, then the norm function is a function $$ \|\cdot\| : X \to \mathbb{R}_{\ge 0} $$ which, when evaluated at $x$, is written $\|x\|$, rather than $\|\,\|(x)$ or $\|\cdot\|(x)$.