Express Norm Using Inner Product

Question

I'd like to know whether there's a way to express a norm using inner product, for example , is there any inner product we may use that is equal to $(||Ax-b||_2)^2$ ? Thanks in advance.

Answer 1

Note that, in your case with the $2$-norm, you have $\|Ax-b\|_2^2 = \langle Ax-b,Ax-b \rangle $, where $\langle \cdot , \cdot \rangle$ is the euclidian scalar product. But this is not always possible, for example the $p$-norm with $1 \leq p < \infty$ defined by $$\|x\|_p := \left(\sum_{k=1}^n |x_k|^p\right)^{1/p},$$ is such that there is no scalar product $\langle \cdot , \cdot \rangle_p$ with $\langle \cdot , \cdot \rangle_p = \|x\|_p$ for $p \neq 2$.

Answer 2

Not in general, for example the infinity norm is a norm on any vector space over a totally ordered field that does not come from an inner product.

E.g. for a vector $\vec{v} = (v_1, \dots, v_n) \in \mathbb{R}^n$, defined $||\vec{v}||_\infty = \max\{v_i\}$. You can check that this is a norm, and does not come from an inner product.