I have been reading Cassels Local Fields and I have a couple of questions regarding valuations that satisfy the triangle inequality.
Cassels defines a valuation on a field $k$ as $|.|:k\to \mathbb{R}$ that is positive definite and multiplicative, together with a constant $C\in\mathbb{R}$ st $|b|\leq 1$ $\implies$ $|1+b|\leq C$.
Cassels also shows that a valuation satisfies the triangle inequality iff we can take $C=2$ in the definition of valuation.
From what I understand:
One can pick any $C$ in the definition of valuation as long as $|b|\leq 1$ $\implies$ $|1+b|\leq C$ is satisfied. Once the choice has been made, $C$ is fixed to check if the valuation indeed satisfies the preceding implication.
There are distinct valuations that satisfy the triangle inequality.
Then, Cassels gives the corollary that 'every valuation is equivalent to the one satisfying the triangle inequality' where one says two valuations $|.|_A,|.|_B$ on $k$ are equivalent if for some fixed $l>0$ real, $|a|_A=|a|_B^l$ for all $a\in k$.
Does this mean that: for every valuation there is a valuation that satisfies the triangle inequality but there need not necessarily be one valuation that satisfies the triangle inequality and is equivalent to all valuations? The way I am imagining this is that (since equivalent valuations belong to an equivalence class) there maybe several distinct equivalence classes of valuations for a given field but each class has a valuation that satisfies the triangle inequality (i.e. where one can take $C=2$).
Lastly, I also believe that not every valuation $|.|$ equivalent to one satisfying the triangle inequality $|.|_t$ itself satisfies the triangle inequality since $|1+b|=|1+b|_t\leq 2^l$ and $l$ could be any large real number.
Any suggestions on why any of this is right or wrong will be hugely appreciated :))
