For questions about vector lattices (aka. Riesz spaces; a type of vector space equipped with a partial order), Banach lattices and similar topics.

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What is the (pre)topology hinted by the Daniell integral

Daniell takes a vector lattice $H$ of a set $\mathbb{R}^X$ which he calls the set of elementary functions. For him, an elementary integral $I$ is a nonnegative functional on $H$ which verifies : if ...
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43 views

Counting sum of lattice points

Assume a set $S$ with $|S|$ entries. Indeed, $S$ is the set of lattice points inside a $k$-sphere. Assume $V=S\oplus S$ where $\oplus$ is the Minkowski sum of two sets. Do you know any lower bound on ...
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53 views

How to obtaining the lattice corresponding to an elliptic curve

Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding ...
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Jordan Decomposition in a Vector Lattice

I'm trying to prove that $x = x^{+} - x^{-}$ for all $x$ in a vector lattice $E$. The relevant definitions: $E$ is a $\bf{\text{vector lattice}}$ if $E$ is a vector space, as well as a partially ...
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124 views

A function that is both log-supermodular and submodular?

Functions that are log-supermodular and submodular do they have some special name? Or any special properties? Any references? Is the function below log-super-modular and sub-modular? Consider ...
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Is the Voronoi region of a lattice symmetric around origion?

Assume an n-dimensional lattice. Is the Voronoi region of the lattice symmetric around origion? In other word, is the following statement true? "if $x\in \mathcal{V}$ then $-x\in \mathcal{V}$" where ...
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191 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
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49 views

Different definitions of Banach lattices?

Do I get it correctly, that the are different definitions of "Banach lattices" available in the literature? To be precise, some authors (like Schaefer) include the order continuity of the norm, while ...
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Computation of determinant of a lattice

Define the following lattice in $\mathbb{Z}^2,$ $$G:=\{\mathbf{x} \in \mathbb{Z}^2:\exists \lambda \in \mathbb{Z} \ \text{such that} \ \mathbb{x}\equiv \lambda (1,1) \mod 5\}. $$ What is the ...
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39 views

do you know any example which is not lattice norm?

A norm $||.||$ on a Riesz space is said to be lattice norm whenever $|x|\leq |y|$ implies $||x||\leq ||y||$. Do you know any example which is not a lattice norm?
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Properties of the Cone of Positive Semidefinite Matrices

The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
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109 views

Planar cross section of leech lattice?

I was wondering what any planar cross sections of the leech lattice would look like. I don't know much about this topic at all, I'm just quite curious. Is there any way to find out?
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92 views

How to `bound' $L^\infty$ by the constant function $1$

Let $(S,\mathcal A, \mu)$ be a measure space and consider the Riesz space $L^\infty=L^\infty(S,\mathcal A, \mu)$ (under point-wise ordering). Let $1_X$ denote the indicator function on $S$ (which is ...
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A question regarding vector spaces with partial order

$\newcommand{\N}{\mathbf{N}}$$\newcommand{\R}{\mathbf{R}}$ Let $X=(X,\leq)$ be a Riesz space (a lattice which is also an ordered vector space over $\R$). Define $X^+=\{x\in X\colon x\ge 0\}$. Are the ...
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Are order continuous Banach lattices with weak unit weakly sequentially complete?

Let $E$ be an order continuous Banach lattice with a weak unit represented as $L^\infty(\mu) \subseteq E\subseteq L^1(\mu)$. Is it true that $E$ is also a weakly sequentially complete Banach lattice? ...
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110 views

completeness of cones in an ordered normed space

Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
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256 views

Show sequence equicontinuous

I don't know how to prove this question: Let $X$ be a compact space, and let $(T_{n})$ be a sequence of positive linear operators on $C(X)$. Also, let $f \in C(X)$ be a strictly positive function. ...
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323 views

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?
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239 views

How does positivity affect operator norms?

Take a measure space $\Omega$, an exponent $1 \le p < \infty$ and a linear operator $T \colon L^p(\Omega) \to L^p(\Omega)$. Suppose that $T$ is positive, that is $Tf \ge 0$ a.e. if $f \ge 0$ almost ...