I've come accross the following inequality for a norm (where the norm defines the length of the vector):
$$\lvert x \rvert ≤ \lvert \lvert x \rvert \rvert \leq \sqrt{n} \lvert x \rvert$$
where $x$ is a vector. Firstly, what is this inequality called? Secondly, in what situation (please also provide an example) does the magnitude of the vector ($\lvert x \rvert$) not equal the norm of the vector ($\lvert \lvert x \rvert \rvert $)?
Thank you!
Best Answer
This is an instance of norm equivalence, here between some norm $\lVert.\rVert$ and the Euclidean norm $\lVert.\rVert_2$. For any two norms $\lVert.\rVert_a$ and $\lVert.\rVert_b$ (of a finite-dimensional vector space) one can give such an equation $$ m \lVert x \rVert_a \le \lVert x \rVert_b \le M \lVert x \rVert_a $$ with specific constants $m$ and $M$.
E.g. if $\lVert.\rVert = \lVert.\rVert_1$ then $\lVert(1,2)\rVert_1 = \lvert 1 \rvert + \lvert 2 \rvert = 3$ and $\lVert(1,2)\rVert_2 = \sqrt{1^2 + 2^2} = \sqrt{5}$.