PlanetMath: topological vector lattice

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topological vector lattice

(Definition)

A topological vector lattice over $\mathbb{R}$ is

a Hausdorff topological vector space over $\mathbb{R}$ ,
a vector lattice, and
locally solid. This means that there is a neighborhood base of 0 consisting of solid sets.

Proposition 1 A topological vector lattice is a topological lattice.

Before proving this, we show the following equivalence on the continuity of various operations on a vector lattice that is also a topological vector space.

Lemma 1 Let be a vector lattice and a topological vector space. The following are equivalent:

$\vee:V^2\to V$ is continuous (simultaneously in both arguments)
$\wedge:V^2\to V$ is continuous (simultaneously in both arguments)
$^+:V\to V$ given by $x^+:=x\vee 0$ is continuous
$^-:V\to V$ given by $x^-:=-x\vee 0$ is continuous
$\vert\cdot\vert:V\to V$ given by $\vert x\vert:=-x\vee x$ is continuous

Proof. $(1\Leftrightarrow 2)$ . If $\vee$ is continuous, then $x\wedge y=x+y-x\vee y$ is continuous too, as

and

are both continuous under a topological vector space. This proof works in reverse too. $(1\Rightarrow 3)$ , $(1\Rightarrow 4)$ , and $(3\Leftrightarrow 4)$ are obvious. To see $(4\Rightarrow 5)$ , we see that $\vert x\vert=x^++x^-$ , since

is continuous,

is continuous also, so that $\vert\cdot\vert$ is continuous. To see $(5\Rightarrow 4)$ , we use the identity

, so that $\vert x\vert=(x+x^-)+x^-$ , which implies $x^-=\frac{1}{2}(\vert x\vert-x)$ is continuous. Finally, $(3\Rightarrow 1)$ is given by $x\vee y=(x-y+y)\vee (0+y)=(x-y)\vee 0+y=(x-y)^++y$ , which is continuous. $\qedsymbol$

In addition, we show an important inequality that is true on any vector lattice:

Lemma 2 Let be a vector lattice. Then $\vert a^+-b^+\vert\le \vert a-b\vert$ for any $a,b\in V$ .

Proof. $\vert a^+-b^+\vert=(b^+-a^+)\vee (a^+-b^+)=(b\vee 0-a\vee 0)\vee(a\vee 0-b\vee 0)$ . Next, $a\vee 0 - b\vee 0 = (b+(-a\wedge 0)\vee (-a\wedge 0)=((b-a)\wedge b)\vee (-a\wedge 0)$ so that $\vert a^+-b^+\vert=((b-a)\wedge b)\vee (-a\wedge 0)\vee ((a-b)\wedge a)\vee (-b\wedge 0)\le (b-a)\vee (-a\wedge 0)\vee (a-b)\vee (-b\wedge 0)$ . Since $(b-a)\vee (a-b)=\vert a-b\vert$ and $a\vee 0$ are both in the positive cone of

, so is their sum, so that $0\le (b-a)\vee (a-b)+(a\vee 0)=(b-a)\vee (a-b)-(-a\wedge 0)$ , which means that $(-a\wedge 0)\le (b-a)\vee (a-b)$ . Similarly, $(-b\wedge 0)\le (b-a)\vee (a-b)$ . Combining these two inequalities, we see that $\vert a^+-b^+\vert\le (b-a)\vee (-a\wedge 0)\vee (a-b)\vee (-b\wedge 0) \le (b-a)\vee (a-b)=\vert a-b\vert$ . $\qedsymbol$

We are now ready to prove the main assertion.

Proof. To show that

is a topological lattice, we need to show that the lattice operations meet $\wedge$ and join $\vee$ are continuous, which, by Lemma 1, is equivalent in showing, say, that

is continuous. Suppose

is a neighborhood base of 0 consisting of solid sets. We prove that

is continuous. This amounts to showing that if

is close to

, then

is close to

, which is the same as saying that if

is in a solid neighborhood

of 0 ( $U\in N$ ), then so is

. Since $x-x_0\in U$ , $\vert x-x_0\vert\in U$ . But $\vert x^+-x_0^+\vert\le \vert x-x_0\vert$ by Lemma 2, and

is solid, $x^+-x_0^+\in U$ as well, and therefore

is continuous. $\qedsymbol$

As a corollary, we have

Proposition 2 A topological vector lattice is an ordered topological vector space.

Proof. All we need to show is that the positive cone is a closed set. But the positive cone is defined as $\lbrace x\mid 0\le x\rbrace = \lbrace x\mid x^-=0\rbrace$ , which is closed since

is continuous, and the positive cone is the inverse image of a singleton, a closed set in $\mathbb{R}$ . $\qedsymbol$

"topological vector lattice" is owned by CWoo.

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Also defines:

locally solid

This object's parent.

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Cross-references: singleton, inverse image, closed, closed set, ordered topological vector space, neighborhood, solid, equivalent, join, meet, lattice, sum, positive cone, inequality, addition, implies, identity, obvious, proof, arguments, continuous, the following are equivalent, operations, equivalence, topological lattice, solid sets, neighborhood base, vector lattice, topological vector space, Hausdorff
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This is version 3 of topological vector lattice, born on 2007-05-09, modified 2007-05-10.
Object id is 9356, canonical name is TopologicalVectorLattice.
Accessed 848 times total.

Classification:

AMS MSC:	06F20 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered abelian groups, Riesz groups, ordered linear spaces)
	46A40 (Functional analysis :: Topological linear spaces and related structures :: Ordered topological linear spaces, vector lattices)

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