What is the meaning of || and F in this equation?

Question

In this chapter on Least Square Problems: http://www4.ncsu.edu/~mtchu/Teaching/Lectures/MA529/chapter4.pdf in (4.2) there is the following formula:

$$g(x_1,\ldots,x_n):=\sum_{i=1}^m\lVert y_i-f(z_i;x_1,\ldots,x_n)\rVert_F^2\tag{4.2}$$

1. They use ||, I thought it should be the absolute of yi-f(...) but I thought that was denoted with a single |, does this mean something else?
2. At the end under the square root they write F, what does this mean?