Let $(E, || \cdot||)$ be a Banach lattice. Let $E_{+}$ denote the positive cone of $E$. The metric $\rho$ on $E_{+}$ is induced by the complete lattice norm $|| \cdot ||$, which is defined by $\rho(x,y) := || x - y||$ for all $x,y \in E_{+}$.

My question is that whether the metric space $(E_{+}, \rho)$ is complete?

I think it is complete. Because we know that a closed subset of Banach space is complete, and the positive cone of a Banach lattice is closed, so that $(E_{+}, \rho)$ is complete.

But I read a textbook wrote a thing that "The metric induced by a complete lattice norm need not be complete". So I am confused now and would like to seek for your help. Thanks in advance!

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    $\begingroup$ It's complete for the reason you wrote. Maybe the textbook says (or would mean to say): "the metric induced by a lattice norm need not be complete", which is correct and means that there exist normed lattices which are not Banach spaces. I think it tries to point out that the lattice property of a norm, although a strong property, it's not strong enough by itself to impose completeness on the underlying space. $\endgroup$ – tree detective Mar 2 '17 at 0:12
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    $\begingroup$ Thank you @treedetective! I got it. Good to see you again:) $\endgroup$ – Paradiesvogel Mar 2 '17 at 0:42

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