Establish Archimedean property of a vector-lattice
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I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem.
I feel the statement below (or in fact weaker versions) should be provable, but I am not being very successful so far. I am posting this question here to get suggestions and/or counterexamples, thanks!
Statement: Let $A$ be a vector lattice (wikipedia entry) and let $Bsubseteq A$ a subset of $A$ which:
1) B is closed under vector spaces operations (of $A$), i.e.: (i) if $b_1,b_2in B$ then $b_1 + b_2 in B$, and (ii) if $bin B$ then $r bin B$ for all $rinmathbb{R}$. In other words, $B$ is a partially orderd vector subspace of $A$, but $B$ is not necessarily a lattice.
2) $B$ generates $A$, i.e., every element $ain A$ is expressible as a finite combination of meet and joins from $B$: $a = bigvee_i bigwedge_j b_{i,j}$.
3) $B$ is Archimedean, in the sense that it does not have infinitesimals (except $0$): for all $b,b^primegeq0$ in $B$, if for all $ninmathbb{N}$ it holds that $nb leq b^prime$, then $b=0$.
Under these assumptions it follows that $A$ is Archimedean as a vector lattice. I.e., for all $a,a^primegeq0$ in $A$, if for all $n$, $n a leq a^prime$ then $a=0$.
end of Statement
As I said, I have not been able to prove this so far. Perhaps there is some counterexample?
functional-analysis vector-spaces lattice-orders vector-lattices banach-lattices
asked Jan 7 at 13:36
makkiatomakkiato
62
$endgroup$
add a comment |
$begingroup$
I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem.
I feel the statement below (or in fact weaker versions) should be provable, but I am not being very successful so far. I am posting this question here to get suggestions and/or counterexamples, thanks!
Statement: Let $A$ be a vector lattice (wikipedia entry) and let $Bsubseteq A$ a subset of $A$ which:
1) B is closed under vector spaces operations (of $A$), i.e.: (i) if $b_1,b_2in B$ then $b_1 + b_2 in B$, and (ii) if $bin B$ then $r bin B$ for all $rinmathbb{R}$. In other words, $B$ is a partially orderd vector subspace of $A$, but $B$ is not necessarily a lattice.
2) $B$ generates $A$, i.e., every element $ain A$ is expressible as a finite combination of meet and joins from $B$: $a = bigvee_i bigwedge_j b_{i,j}$.
3) $B$ is Archimedean, in the sense that it does not have infinitesimals (except $0$): for all $b,b^primegeq0$ in $B$, if for all $ninmathbb{N}$ it holds that $nb leq b^prime$, then $b=0$.
Under these assumptions it follows that $A$ is Archimedean as a vector lattice. I.e., for all $a,a^primegeq0$ in $A$, if for all $n$, $n a leq a^prime$ then $a=0$.
end of Statement
As I said, I have not been able to prove this so far. Perhaps there is some counterexample?
functional-analysis vector-spaces lattice-orders vector-lattices banach-lattices
asked Jan 7 at 13:36
makkiatomakkiato
62
$endgroup$
$begingroup$
Do you have an example motivating this question?
$endgroup$
– user458276
Mar 16 at 22:41
$begingroup$
Any examples would need to be infinite dimensional (see math.leidenuniv.nl/scripties/MasterJongen.pdf). Also, since you’d have to a finite combination of meets and joins of elements of $B$ to produce any elements of $A$, it seems unlikely that there are any finite proper subsets $B$ that satisfy (2).
$endgroup$
– user458276
Mar 17 at 0:41
add a comment |
$begingroup$
I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem.
I feel the statement below (or in fact weaker versions) should be provable, but I am not being very successful so far. I am posting this question here to get suggestions and/or counterexamples, thanks!
Statement: Let $A$ be a vector lattice (wikipedia entry) and let $Bsubseteq A$ a subset of $A$ which:
1) B is closed under vector spaces operations (of $A$), i.e.: (i) if $b_1,b_2in B$ then $b_1 + b_2 in B$, and (ii) if $bin B$ then $r bin B$ for all $rinmathbb{R}$. In other words, $B$ is a partially orderd vector subspace of $A$, but $B$ is not necessarily a lattice.
2) $B$ generates $A$, i.e., every element $ain A$ is expressible as a finite combination of meet and joins from $B$: $a = bigvee_i bigwedge_j b_{i,j}$.
3) $B$ is Archimedean, in the sense that it does not have infinitesimals (except $0$): for all $b,b^primegeq0$ in $B$, if for all $ninmathbb{N}$ it holds that $nb leq b^prime$, then $b=0$.
Under these assumptions it follows that $A$ is Archimedean as a vector lattice. I.e., for all $a,a^primegeq0$ in $A$, if for all $n$, $n a leq a^prime$ then $a=0$.
end of Statement
As I said, I have not been able to prove this so far. Perhaps there is some counterexample?
functional-analysis vector-spaces lattice-orders vector-lattices banach-lattices
asked Jan 7 at 13:36
makkiatomakkiato
62
$endgroup$
I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem.
I feel the statement below (or in fact weaker versions) should be provable, but I am not being very successful so far. I am posting this question here to get suggestions and/or counterexamples, thanks!
Statement: Let $A$ be a vector lattice (wikipedia entry) and let $Bsubseteq A$ a subset of $A$ which:
1) B is closed under vector spaces operations (of $A$), i.e.: (i) if $b_1,b_2in B$ then $b_1 + b_2 in B$, and (ii) if $bin B$ then $r bin B$ for all $rinmathbb{R}$. In other words, $B$ is a partially orderd vector subspace of $A$, but $B$ is not necessarily a lattice.
2) $B$ generates $A$, i.e., every element $ain A$ is expressible as a finite combination of meet and joins from $B$: $a = bigvee_i bigwedge_j b_{i,j}$.
3) $B$ is Archimedean, in the sense that it does not have infinitesimals (except $0$): for all $b,b^primegeq0$ in $B$, if for all $ninmathbb{N}$ it holds that $nb leq b^prime$, then $b=0$.
Under these assumptions it follows that $A$ is Archimedean as a vector lattice. I.e., for all $a,a^primegeq0$ in $A$, if for all $n$, $n a leq a^prime$ then $a=0$.
end of Statement
As I said, I have not been able to prove this so far. Perhaps there is some counterexample?
functional-analysis vector-spaces lattice-orders vector-lattices banach-lattices
functional-analysis vector-spaces lattice-orders vector-lattices banach-lattices
asked Jan 7 at 13:36
makkiatomakkiato
62
asked Jan 7 at 13:36
makkiatomakkiato
62
asked Jan 7 at 13:36
makkiatomakkiato
62
asked Jan 7 at 13:36
makkiatomakkiato
62
asked Jan 7 at 13:36
makkiatomakkiato
62
62
$begingroup$
Do you have an example motivating this question?
$endgroup$
– user458276
Mar 16 at 22:41
$begingroup$
Any examples would need to be infinite dimensional (see math.leidenuniv.nl/scripties/MasterJongen.pdf). Also, since you’d have to a finite combination of meets and joins of elements of $B$ to produce any elements of $A$, it seems unlikely that there are any finite proper subsets $B$ that satisfy (2).
$endgroup$
– user458276
Mar 17 at 0:41
add a comment |
$begingroup$
Do you have an example motivating this question?
$endgroup$
– user458276
Mar 16 at 22:41
$begingroup$
Any examples would need to be infinite dimensional (see math.leidenuniv.nl/scripties/MasterJongen.pdf). Also, since you’d have to a finite combination of meets and joins of elements of $B$ to produce any elements of $A$, it seems unlikely that there are any finite proper subsets $B$ that satisfy (2).
$endgroup$
– user458276
Mar 17 at 0:41
$begingroup$
Do you have an example motivating this question?
$endgroup$
– user458276
Mar 16 at 22:41
$begingroup$
Do you have an example motivating this question?
$endgroup$
– user458276
Mar 16 at 22:41
$begingroup$
Any examples would need to be infinite dimensional (see math.leidenuniv.nl/scripties/MasterJongen.pdf). Also, since you’d have to a finite combination of meets and joins of elements of $B$ to produce any elements of $A$, it seems unlikely that there are any finite proper subsets $B$ that satisfy (2).
$endgroup$
– user458276
Mar 17 at 0:41
$begingroup$
Any examples would need to be infinite dimensional (see math.leidenuniv.nl/scripties/MasterJongen.pdf). Also, since you’d have to a finite combination of meets and joins of elements of $B$ to produce any elements of $A$, it seems unlikely that there are any finite proper subsets $B$ that satisfy (2).
$endgroup$
– user458276
Mar 17 at 0:41
add a comment |
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$begingroup$
Do you have an example motivating this question?
$endgroup$
– user458276
Mar 16 at 22:41
$begingroup$
Any examples would need to be infinite dimensional (see math.leidenuniv.nl/scripties/MasterJongen.pdf). Also, since you’d have to a finite combination of meets and joins of elements of $B$ to produce any elements of $A$, it seems unlikely that there are any finite proper subsets $B$ that satisfy (2).
$endgroup$
– user458276
Mar 17 at 0:41