Negative of a power of a norm.

Question

I have one silly doubt if $\|.\|$ is a norm on a Hilbert space, then is it correct that $$\|x\|^{\mu} = \|-x\|^{\mu}, \qquad \mu \in (0, 1)$$ Please help me to understand the above concept.

According to me, it must be correct as $\|\alpha x \| = |\alpha|\|x\|$, for any scaler $\alpha$.

Answer 1

We have for a norm $\|.\|$ in a Hilbert-space and $\mu\in(0,1)$ \begin{align*} \color{blue}{\|-x\|^{\mu}}=\|(-1)\cdot x\|^{\mu}=\left(|-1|\|x\|\right)^{\mu}=\color{blue}{\|x\|^{\mu}} \end{align*}

Answer 2

By definition of a norm (see https://en.wikipedia.org/wiki/Norm_(mathematics)#Definition) it is absolutely scalable as you wrote $\|\alpha x\|=|\alpha|\cdot\|x\|$. Therefore it is true to conclude $\|-x\|=\|x\|$. So, this holds even in normed spaces and further the exponent $\mu$ is totally irrelevant.