What is the $\lVert$ symbol?

Question

I am trying to understand the quadratic equation below but cannot understand what the double bars stand for. $$\lVert W_L LP' \rVert^2 + \sum_i W_{H,i}^2 \lVert p_i' - p_i\rVert$$

Answer 1

Most likely the double lines is what is called a norm. We for example the standard norm $\lvert \cdot \lvert$ on the real numbers $\mathbb{R}$ defined by $$ \lvert x \lvert =\begin{cases}x & \text{if } x \geq 0 \\ -x & \text{if }x < 0 \end{cases}. $$ Or you have the norm (also denoted $\lvert \cdot \lvert$) on the complex number $\mathbb{C}$ defined by $\lvert a + ib\lvert^ = \sqrt{a^2 + b^2}$.

Note that both of these norms satisfy certain properties: $$\begin{align} &(1)\quad \lvert xy\lvert = \lvert x\lvert \lvert y \lvert \\ &(2)\quad \lvert x + y\lvert \leq \lvert x\lvert + \lvert y \lvert \\ &(3)\quad \lvert x\lvert = 0 \iff x = 0. \end{align} $$

Now in general if you have a vector space $V$ over the field of complex numbers $\mathbb{C}$, you can talk about an (abstract) norm as being any function $\lVert \cdot \lVert : V \to [0, \infty)$ satisfying basically the three conditions above: $$\begin{align} &(1)\quad \lVert \lambda x\lVert = \lvert \lambda \lvert \lVert x \lVert \\ &(2)\quad \lVert x + y\lVert \leq \lVert x\lVert + \lVert y \lVert \\ &(3)\quad \lVert x\lVert = 0 \iff x = 0. \end{align} $$ Here $\lambda\in \mathbb{C}$ and $x,y \in V$, and $\lvert \lambda \lvert$ is the norm on $\mathbb{C}$.

So some tend to use the double lines $\lVert$ for norms other than those on the real or the complex numbers, but it is really just a matter of notation.