what does the notation $\|x\|$ where x is a vector mean?

Question

I have come across something that requires minimizing $\|x\|$ where $x$ is a vector? What does the notation $\|x\|$ mean? Is it the two norm or something else?

Answer 1

$\lVert . \rVert$ is a norm. There are defininitions for vectors, e.g. $\lVert x \rVert$, or matrices $\lVert A \rVert$.

If nothing extra is said, you can assume the Eucledian norm.

Oterwise subscripts are common, e.g. $$ \lVert x \rVert_p = \left(\sum_i x_i^p \right)^{1/p} $$

On the other hand, thanks to norm-equivalence, the specific norm might not matter for some use cases.