What is meant by the double vertical line notation here?

Question

What do the double vertical lines around $\vec i$ and $\vec j$ in this equation actually mean?

$$ sim(i,j) = cos(\vec i, \vec j) = \frac{\vec i \cdot \vec j}{\lVert \vec i\rVert^2 * \lVert\vec j\rVert^2} $$

Answer 1

It signifies the "norm" of the vector. There are many types of norms you can define, which all follow basic rules. Based on what you posted, they are using it to refer to the "Euclidean Norm" which is, as the comments suggest, the "length" of the vectors here. Single lines always means the Euclidean Norm (from my experience) but double lines can mean any norm.

Answer 2

When discussing real or complex vectors, you often also have real or complex numbers in the conversation. If these things are represented symbolically, every little thing you can do to remind the reader which symbols represent vectors and which symbols represent numbers is good. For example, the meaning of $\left|a\right|\left|v\right|$ is not as clear as that of $\left|a\right|\left|\vec{v}\right|$, and it's even more clear to write $\left|a\right|\left\|\vec{v}\right\|$.

Incidentally, \left\|...\right\| looks better than \mid\mid...\mid\mid. Compare $\left\|\vec{v}\right\|$ to $\mid\mid\vec{v}\mid\mid$.