Hello!!! 
Let \(\displaystyle c \in \mathbb{R}, 0<c<1\) and \(\displaystyle p \in \mathbb{P}\). We consider the function \(\displaystyle \theta_p:\mathbb{Q}\rightarrow \mathbb{R}\)
\(\displaystyle x=p^{w(x)}u \mapsto c^{w(x)}\).
Show that \(\displaystyle \theta_p\) is a p-Norm.
According to my notes:
A p-norm of \(\displaystyle \mathbb{Q}_p\) is a function \(\displaystyle ||_p: \mathbb{Q}_p \to \mathbb{R}\).
\(\displaystyle x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}\)
\(\displaystyle \text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0\)
Identities:
-\(\displaystyle |x|_p \geq 0 \text{ and } |x|_p=0 \Leftrightarrow x=0\)
- \(\displaystyle |xy|_p=|x|_p |y|_p\)
-\(\displaystyle |x+y|_p \leq \max \{ |x|_p, |y|_p\} \\ \leq |x|_p+|y|_p\)
If \(\displaystyle |x|_p \neq |y|_p \Rightarrow |x+y|_p=\max \{ |x|_p, |y|_p\}\)
-\(\displaystyle \mathbb{Z}_p=\{ x \in \mathbb{Q}_p | |x|_p \leq 1 \}\)
-\(\displaystyle \mathbb{Z}_{p}^*=\{ x \in \mathbb{Z}_p | |x|_p=1 \}\)
In order to show that \(\displaystyle \theta_p\) is a p-norm, do I have to show that it satisfies all of the above five identities?
Let \(\displaystyle c \in \mathbb{R}, 0<c<1\) and \(\displaystyle p \in \mathbb{P}\). We consider the function \(\displaystyle \theta_p:\mathbb{Q}\rightarrow \mathbb{R}\)
\(\displaystyle x=p^{w(x)}u \mapsto c^{w(x)}\).
Show that \(\displaystyle \theta_p\) is a p-Norm.
According to my notes:
A p-norm of \(\displaystyle \mathbb{Q}_p\) is a function \(\displaystyle ||_p: \mathbb{Q}_p \to \mathbb{R}\).
\(\displaystyle x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}\)
\(\displaystyle \text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0\)
Identities:
-\(\displaystyle |x|_p \geq 0 \text{ and } |x|_p=0 \Leftrightarrow x=0\)
- \(\displaystyle |xy|_p=|x|_p |y|_p\)
-\(\displaystyle |x+y|_p \leq \max \{ |x|_p, |y|_p\} \\ \leq |x|_p+|y|_p\)
If \(\displaystyle |x|_p \neq |y|_p \Rightarrow |x+y|_p=\max \{ |x|_p, |y|_p\}\)
-\(\displaystyle \mathbb{Z}_p=\{ x \in \mathbb{Q}_p | |x|_p \leq 1 \}\)
-\(\displaystyle \mathbb{Z}_{p}^*=\{ x \in \mathbb{Z}_p | |x|_p=1 \}\)
In order to show that \(\displaystyle \theta_p\) is a p-norm, do I have to show that it satisfies all of the above five identities?