The meaning of double slashes and subscripts of a function

Question

I was asked to find a derivative of this function $$ f(x)=\frac{1}{2}\|Ax-b\|_2^2+\frac{\gamma}{2}\|x\|_2^2. $$

The problem was I didn't know the meanings of the double vertical slashes $\|\cdot\|$ and the top and bottom subscripts 2 at the end of the ||. From my own research, ||v||2 means the norm of vector v in 2 dimensions. But that still didn't help me understand the meaning of the whole function expression above.

Could you please help explain the meaning of the above function and how to find its derivative? Thanks in advance.

Answer 1

$|\hspace{-1pt}|\cdot|\hspace{-1pt}|$ simply means the norm. For, $p\geq1$, the $p$-norm is defined as $|\hspace{-1pt}|(x_1,\dots,x_n)|\hspace{-1pt}|_p=\sqrt[p]{|x_1|^p+\dots+|x_n|^p}$. Hence, $|\hspace{-1pt}|(x_1,\dots,x_n)|\hspace{-1pt}|_p^p$ simply means $|x_1|^p+\dots+|x_n|^p$. In particular, $p=2$ referes to the Euclidean norm as $$|\hspace{-1pt}|(x_1,\dots,x_n)|\hspace{-1pt}|_2^2=|x_1|^2+\dots+|x_n|^2$$

Answer 2

The vertical lines represent a norm. The subscript $2$ tells us it's the Euclidean norm which is a p-norm. It's basically the length of the vector.