Example- $l_p$ norm space

Question

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers).

To prove that above is norm on $l_p$ space; firstly, we need to show that $||x||\ge0$.[first property of norm space]

It is written in my book that this is trivially. But I couldn't convince myself.

Here $x=<x_1,x_2........>$ sequences, where scalars $x_i\in \mathfrak C$, So $|x_i|>0$ and their sum is positive. But its whole power is $1/p$ then how come its always positive. for example if $p= 1/2$ then $||x||$ has two values one is positive and another negative.

Answer 1

In this context (and quite generally in the context of analysis) $x^\alpha $ for non-negative $x$ is assumed to be the non-negative value of $x^\alpha $ (in case more than one value is possible). So, in particular, $x^{1/2}$ means the non-negative square root of $x$.

Answer 2

Let $x=(x_1,x_2,x_3,...)\in \ell_p$, then as $|x_i|\geq 0$, $\forall i$, we have that $|x_i|^p\geq 0$, $\forall i$ and $\forall p$, then $\sum |x_i|^p \geq 0$ and so $$||x||=\left(\sum_{i=1}^{\infty}|x_i|^p\right)^{\frac{1}{p}}\geq 0. $$