Htykuut

I'm reading a paper on a Banach space with an order induced by a cone.

Let $(mathbb{X}, || cdot||)$ be a real Banach space.

We define a subset $P$ of $mathbb{X}$ by $P := { x in mathbb{X} : xgeq 0}$.

My question is what does $ xgeq 0$ mean, knowing that $mathbb{X}$ is quite abstract and could be any real Banach space.

Given an open cone $K$ in a Banach space (i.e. a subset closed with respect to summation and multiplication by non negative factors $lambdageq 0$, and with zero intersection with it's opposite $-Kcap K = {0}$) you can define the order relation by $a>b$ iff $a-bin K$.

In an analogous way, given a closed cone $C$ you can define the order relation by $ageq b$ iff $a-bin C$.

So if you assume the cone to be closed then you'll simply have $P=C$, otherwhise $P$ will be the closure of $K$.

To rephrase it to answer your question: $xgeq 0$ means for $x$ to be in the closure of the cone you use to define your partial order on $mathbb{X}$.

Edit:

Note that we'd like to define a continuous partal order (i.e. such that if $ageq b_lambda$ for each element of a net ${b_lambda}_{lambdainLambda}$ converging to $b$, then $ageq b$), and in order to have this property, you need your defining cone to be closed.