I am struggling with understanding the existence of the supremum.
$L$ is a relatively uniformly complete Riesz space. $L+iL=\{φ:φ=f+ig, f,g∈L\}$ is a complexification of $L$.
Modulus in the vector space complexification is $|φ|=|f+ig|=\sup\{f\times cos(θ)+g\times sin(θ), 0≤θ≤2π\}.$
Show that supremum on the right hand side exists.
Put
$h(θ)=|fcos(θ)+gsin(θ)|\ 0≤θ≤2π$ and $e=|f|+|g|$
$h_n=\sup\{h(2π.k.2^{-n});k=0,1,...2^n\} n=0,1,2... $
Using that for all $f_1,...,f_n,g_1,...,g_n$ in L
we have $|sup(f_1,...,f_n)-sup(g_1,...,g_n)|≤sup(|f_1-g_1|,...,sup|f_n-g_n|) $
which is a sharper version of Birkhoff inequality. It is not hard to see that $h_n$ is an increasing $e$- uniform Cauchy sequence.
I couldn't understand that $h_n$ is an increasing $e$-uniform Cauchy sequence.
