# Questions tagged [banach-lattices]

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

28 questions
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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 $V^+$ ...
1answer
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### Weakly null sequence in Banach lattices

Let $(x_n)_n$ be a positive, disjoint, weakly null sequence in a Banach lattice $E$. If $(y_n)_n$ is a sequence such that $0\leq y_n\leq x_n$ for every $n\in \mathbb{N}$, we can garantee that $y_n$ is ...
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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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### 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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### 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 ...
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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 ...
2answers
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### Infinite dimensional Banach lattice $L^\infty(X)$ is not order continuous

Consider an arbitrary measure space $(X,\Sigma,\mu)$, with the only assumption being that $L^\infty(X)$ is infinite dimensional. Consider $L^\infty(X)$ as a Banach lattice with the usual ordering. As ...
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### Hardy space as a Banach lattice

The Hardy spaces $H^p$ of holomorphic functions on the unit disk are Banach spaces. Question: Are they also Banach lattices? If yes, why is it less common to consider the Hardy spaces as Banach ...
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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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### 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 ...
1answer
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### “Representation Capacity” of Finite Lattice Ordered Modules

Apologies for what is probably a very basic question, but I am looking for references for what sort of lattices can be represented in what I think should be called something like "finite lattice-...
1answer
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### Positive cone of Banach lattice algebra

From the literature on Banach lattice algebras (and that on ordered Banach algebras) there does not appear to be a consensus on a definition. What is agreed is that one should be a Banach lattice, be ...
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### Division in Banach Lattice Algebra

Let X be a Banach Lattice Algebra and $X_+=\{f\in X: f>0\}$. Let $f:X_+\rightarrow X$ be continuously differentiable. Question: When does the expression $\frac{f'(x)}{x}$ for $x\in X_+$, make ...
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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 ...
1answer
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### Are dual spaces with unconditional bases weakly sequentially complete?

It is well known that a weakly sequentially complete Banach space with an unconditional basis is isomorphic to a conjugate space. Is the converse to this statement true? If a Banach space is a ...
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### C*-algebras as Banach lattices?

It seems to be trivial but I am not sure about monotonicity of the norm in the non-commutative case: Is every C*-algebra a Banach lattice with respect to its natural positive cone?