2-normed square meaning

Question

I heard $2$-normed square in a lecture talking about the objective function of least-squares. What does the $2$ mean? I understand we take norm and square it, $2$ doesn't make sense to me. $$\|Ax−B\|^2_2$$ In the above equation, I am referring to the subscript.

Answer 1

This is probably referring to the case $p=2$ of a $p$-norm.

Answer 2

The $2$ in the subscript means that it's the two-norm or euclidean norm, meaning that if we have, for example, an $x=(x_1,x_2,\ldots,x_n)$, then: $$\| x \|_2^2=\sqrt{\sum^n_{i=1} |x_i|^2}^2$$

The two norm is a special case of the $p$-norm so $\| x\|_p^2$ becomes $$\| x\|_p^2 = \sqrt[p]{\sum_{i=1}^n |x_i|^p }^2.$$