# Tagged Questions

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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### Gram-Schmidt process - Division by zero (ERROR)

I'm working with full-rank lattice basis, and I need to compute the Gram-Schmidt norms and coefficients to measure its quality. But during the process I have a division by 0. The division by zero is ...
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### Show that $\mathcal{F}$ is a lattice fulfilling Stone's Axiom

Consider $$f^+(0):=\lim_{r\searrow 0}\frac{f(r)}{r},~~~~~~\mathcal{F}:=\left\{f\in C([0,1],\mathbb{R}): f(0)=0, f^+(0)\text{ exists}\right\}.$$ Moreover, let $\mathcal{F}^+$ be the set of all non-...
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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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### In a Riesz space, does $(u+v)^+=u^+ + v^+$ hold?

Assume that $E$ is a Riesz space (lattice ordered vector space). For $u\in E$, let $u^+ = u\vee 0$. Then, for $u,v\in E$, does $(u+v)^+=u^+ + v^+$ hold?
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### Closed sublattice generated by countable set

Apparently the closed sublattice generated by a countable subset of a Banach lattice is separable. I am trying proof this, but I am stuck, does anybody have an idea? Trying to prove the above, I ...
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### Orientation Preserving Lattice Equivalence

I have been given the following problem: Given a vertex $x$ of a polygon P, show there is a unique orientation preserving lattice equivalence $\Phi$ so that (i) $\Phi(x)=(0,0)^t$ (ii) There are co-...
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### finding the basis of a kernel in a lattice

Given a parity check matrix $A$ we define the $q$-ary lattice $$\Lambda(A) = \{x \in \mathbb Z^m\;:\;Ax\equiv0\pmod q\}$$ How to find the basis of the lattice and how to find its hermite normal form?
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### Basic properties of Riesz spaces

I'm self studyhing from Peter Meyer-Nieberg's Banach Lattices, and I'm having some trouble with some of the very basic properties. So, what I have to work with at this point is the definition: We ...
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### When is a lattice dense in a torus?

Let $\pi: \mathbb{R}^n\rightarrow \mathbb{R}^n/\mathbb{Z}^n$. What (necessary and sufficient) criteria on $A\in GL_n(\mathbb{R})$ guarantee $\pi(A\mathbb{Z}^n)$ is dense?
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### What is an ideal lattice?

Can anyone describe in layman's terms what an ideal lattice is? I've seen them mentioned in many places, but haven't found a good definition of what exactly they are, nor any good terms to know where ...
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### What does the topology on $SL(n, \mathbb{R}) / SL(n, \mathbb{Z})$ intuitively look like?

I have come across Mahler's compactness criterion, and am having trouble wrapping my head around the topology of the moduli space of unit volume lattices. Is there an intuitive way to think about it, ...
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### Creating n-dimensional lattices from lower dimensional parts

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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### How do I calculate the area the Wigner Seitz cells cover in a square?

It's my first time here, so I appologise in advance if I break any rules through this post. So I have a Cartesian Lattice spanning across the Euclidean plane and a unit square. The lattice points ...
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### Set of Riesz homomorphisms

In a text I am using it states the following: "The set of all Riesz homomorphisms between two Riesz spaces does not ordinarily have a simple structure of its own. Consider for example, the set of ...
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### n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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### Definitions of order-dense Riesz spaces

I would like to get an intuitive idea as to why the definitions of order-dense and locally order-dense Riesz subspaces were chosen in the following way: A Riesz subspace $F$ of $E$ is order-dense if ...
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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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### Is there a connection between lattices in the sense of orders and lattices in the sense of groups?

I'm wondering about this for some time now - is there a intuitive connection between those concepts or have they been named the same by chance? In particular, my interest in this question was ...
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### 6Sz as the automorphism group of the complex Leech lattice

Consider the Leech lattice as a complex lattice over the Eisenstein integers. Both in Conway ("Sphere packings, lattices and groups") and Wilson ("The Complex Leech Lattice and Maximal Subgroups of ...
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### invariance of angles under lattice automorphisms

Let $L$ be a lattice, $x,y\in L$ and $g\in Aut(L)$. Does this imply $$<x,y>=<g(x),g(y)>,$$ in other words, are angles or inner products between any two lattice vectors invariant under ...
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### Lattice generated by vectors orthogonal to an integer vector

Given a non-zero vector $\boldsymbol{v}$ composed of integers, imagine the set of all non-zero integer vectors $\boldsymbol{u}$, such that $\boldsymbol{u} \cdot \boldsymbol{v} = 0$, i.e., the integer ...
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