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Questions tagged [banach-lattices]

Banach lattices are Banach spaces endowed with a partial ordering that is compatible with the norm.

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Is a closed subset of a Banach lattice complete?

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,...
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Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in $E^{**}$ ...
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Irreducible operators

I would like some reference to books that talk about irreducible operators on Banach lattices and its properties. A quick google search did not reap fruitful results. Edit 1: I'm especially ...
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Show that two projections commute if and only if their respectively closed subspaces are compatibles

I have the next problem: Let $\mathcal{H}$ be a separable and complex Hilbert space, with $S$ and $Q$ two closed subspaces on it. Let $P_S$ and $P_Q$ be the orthogonal projection onto $S$ and $Q$ ...
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Establish Archimedean property of a vector-lattice

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 ...
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Reference for Banach Lattices

I am looking for a survey or a basic book on Banach Lattices (I know next to nothing about the subject, so a basic survey is what I believe will be helpful). Can somebody please suggest any good ...
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Closed subalgebras of $C[0,1]$ and $C[0,1)$?

Let $C[0,1]$ be the algebra of continuous functions $f: [0,1] \to \mathbb R$. Let $C[0,1)$ be the algebra of bounded continuous functions $f: [0,1) \to \mathbb R$. Can anyone give some interesting ...
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$x'_n\overset{w^\ast}{\to}0$ and $x_n\overset{w}{\to}0$ implies $\sup_m |x_m'|(|x_n|)\to 0$?

If $x_n'$ is weak-$\ast$-ly null and $x_n$ is weakly null in a Banach lattice, do we always have $$\sup_m |x_m'|(|x_n|)\to 0\,?$$ Thank you! Background: Definition weak(-$\ast$) topology: On the ...
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Strong convergence of regular operators and convergence of the modulus

Let $E$ be a Banach lattice and $T_n\in\mathcal{L}^r(E)$ a sequence of regular operators such that $T_n$ converges strongly to $T\in\mathcal{L}^r(E)$. How to prove that $\left|T_n\right|$ converges ...
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A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
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The ideal generated by a maximal orthogonal system in a Banach lattice

I have a pretty specific question about H.H. Schaefer's "Banach lattices and positive operators" book. In chapter 3, part 6 (page 169), it is said that the ideal generated by a set S (which is a ...
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$s$-convexity implies $p$-convexity in Banach lattices

If $1<p<s<\infty$ and $E$ is a Banach lattice which is $s$-convex, is it also $p$-convex? If so, what would be a good reference for this?
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Bounded set and norm bounded set in a Banach lattice space

I am reading about Banach lattice space and confuse a little bit about two concepts "bounded" and "norm bounded set". Could you please help me to declare them? More precisely, let $E$ be a Banach ...
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Proof of additivity of the positive linear functional $\phi^+$ on a vector/Banach lattice that will be $\phi\vee0$.

For context, this is used in defining $\phi\vee0$ in a proof that the dual of a vector lattice is a vector lattice. Given a linear functional $\phi$ on a vector lattice $V$, define $\phi^+$ on $V^+$ ...
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closed lattice ideal is isomorphic to $C(K)$

Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is ...
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Unbounded norm convergence

I study on unbounded norm convergence in Banach lattice. I have to find an example.Is there any example of unbounded norm convergence but not norm convergence in Banach lattice?
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$(V, \geq)$ Banach lattice, $(W, \geq)$ Riesz space, then $\mathcal{L}(V, W)^+ = \mathcal{B}(V, W)^+$

Let $(V, \geq)$ be a Banach lattice and $(W, \geq)$ be Riesz space whose whose positive cone is generated by some positive element of the space. Then $\mathcal{L}(V, W)^+ = \mathcal{B}(V, W)^+$, where ...