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

learn more… | top users | synonyms (1)

1
vote
1answer
24 views

Show space of C1 functions on (0,1) is a Banach lattice

I'm working on the beginning of a book on Banach lattices, and it wants me to show that $C^1(0,1)$ is a Banach Lattice, with the norm $\|f\|=\|f'\|_\infty+|f(0)|$ and the order $f\le g$ if $f(0)\le f(...
1
vote
0answers
19 views

Characterization of Banach sublattice of L^1

Let $(X, \Sigma, \mu)$ be a measure space and let $F\subset L^1(X,\Sigma,\mu)$ be a Banach sublattice of $L^1$ with the following properties: (1) If $f\in F$, $f$ real-valued, then $f\land 1\in F$ (...
0
votes
1answer
15 views

Vector lattice: $[0,v]+[0,u] = [0,u+v]$

Let V be a real vector-lattice and $u\geq 0, v\geq 0$. I need to show that $[0, u+v] \subseteq [0,u]+[0,v]$. I would appreciate a hint on how to start. For $z\in [0,u+v]$ with $z\geq u$ or $z\...
0
votes
0answers
37 views

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 ...
1
vote
0answers
28 views

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-...
4
votes
0answers
34 views

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 $E^{**}$...
1
vote
0answers
36 views

Show that minimum and maximum are contained in $V(\mathcal{R})$/ Stones' axiom

Let $\mathcal{R}\subset\mathcal{P}(\Omega)$ be a ring for some set $\Omega$. Consider $$ V:=V(\mathcal{R}):=\left\{\sum_{i=1}^n\alpha_i1_{A_i}: \alpha_i\in\mathbb{R}, A_i\in\mathcal{R}, n\in\...
0
votes
0answers
16 views

Adjoint lattice homomorphic if surjective

Is the adjoint of a linear operator $T:X\to Y$ between Banach lattices, always lattice homomorphic if $T$ is surjective? This is my proof but I really doubt this is true: $\forall a,b\in Y'$ and $\...
2
votes
1answer
23 views

Are all adjoints lattice homomorphisms?

Obviously something must be wrong in the following reasoning proving that any linear operator $T:X\to Y$ between Banach lattices has a lattice homomorphic adjoint: $\forall a,b\in E':$ $$T'(a\wedge b)=...
0
votes
1answer
15 views

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?
1
vote
0answers
10 views

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 ...
0
votes
0answers
10 views

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-...
0
votes
0answers
57 views

Definition of q-ary lattices

Lattices defined over the vector space over $\mathbb{R}^n$, whereas q-ary lattices consists of only integers i.e., Let A be a $\mathbb{Z}_q^{n\times m}$ then q-ary lattice is defined as $$\Lambda(A) = ...
0
votes
0answers
24 views

Smoothing parameter in lattices

Two bounds are defined for smoothing parameter. The first one is related to the minimum distance of its dual lattice. The second bound is on Gram-Schmidt minimum which is given by $$\widetilde{bl} = ...
0
votes
0answers
11 views

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?
2
votes
1answer
25 views

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 ...
3
votes
1answer
88 views

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?
0
votes
0answers
15 views

Real measures inequality

I'm interested in this textbook exercise about complex measures: We have real measures (measures taking values in finite real numbers) $\mu$, $\nu$ and $\lambda$, with the following relation: $$\...
0
votes
1answer
54 views

Is the supremum of the closure equal to the supremum of the set?

Let $X$ be any Banach space and $M\subset X$ be bounded. We know the $\sup(M)\in\overline{M}$ in general. Since $M$ is bounded $\sup_{u\in M}\|u\|<\infty$. Question: Can we somehow write that $$\...
1
vote
1answer
51 views

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 ...
5
votes
2answers
54 views

Compatible Hilbert space subspaces - need help understanding a statement made in a book

A book I'm reading has the following in a section on lattices formed by subspaces of a Hilbert space : Two subspaces $M$ and $N$ are compatible if there exist three mutually disjoint subspaces $...
0
votes
1answer
18 views

Bases and superbases

So, in the book "The Sensual (Quadratic) Form" By Conway and Fung, together with the notion of the base for the lattice, there is also a notion of superbase: "In the same spirit, a strict base is an ...
0
votes
1answer
29 views

Does positive part in subspace which is Riesz space equal positive part in the full Riesz space

In my thesis I encountered the following problem: suppose that $E,F$ are Riesz spaces such that $E$ is a subspace of $F$ and that the ordering on $E$ matches the one on $F$, i.e. $x\leq_E y\Rightarrow ...
2
votes
1answer
32 views

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, ...
1
vote
0answers
38 views

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 ...
1
vote
1answer
3k views

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 ...
1
vote
1answer
25 views

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 ...
0
votes
0answers
46 views

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 ...
4
votes
1answer
54 views

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 ...
1
vote
0answers
25 views

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 ...
1
vote
1answer
41 views

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 ...
2
votes
0answers
26 views

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 ...
0
votes
0answers
22 views

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 ...
0
votes
1answer
41 views

Question about an extended linear functional on vector lattice of functions

I'm struggling with following problem: Let $\mathcal{F}$ be a vector lattice of bounded functions on a set $X$ such that $1\in\mathcal{F}$. Suppose that we are given a linear functional $L$ on $\...
0
votes
1answer
25 views

Sacle the distance of lattice points

I know that for a hexagonal lattice generated by (0,1) and ($\sqrt{3}/2$,1/2) (i.e., when the distance between lattice points is 1), the number of lattice points in a circle of given radius $r$ can be ...
4
votes
1answer
44 views

How to find almost periodic lattices in a set of high-dimensional points?

sorry for lame question, but I just have no maturity in this direction. Let's say I have very large set ( millions ) of high-dimensional vectors ( typical dimensionality is 64). These vectors ...
4
votes
1answer
36 views

What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
0
votes
0answers
25 views

inverse relation in a vector lattice

Let $X$ be a vector lattice endowed with an ordering $\leq$. If for some operator $\phi:X_+\rightarrow X$ and $m,M\in X$, we have $$m\leq \phi(x)\leq M,\quad x\in X_+. $$ What would be the bound for ...
1
vote
1answer
33 views

Error in my exercise concerning Riesz spaces and Yosida's Lemma.

I was given this exercise in class today: Using Yosida's Lemma, proof that there exists a Riesz-homomorphism $\varphi: BC(\mathbb{R}) \to \mathbb{R}$ such that $\varphi(\textbf{1}) = 1$ and $\...
3
votes
1answer
107 views

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 ...
2
votes
0answers
69 views

Leech Lattice and Golay Code

Consider the following Miracle Octat Generator or MOG. Choose the sign $\pm 3$ and fill in the blanks $\pm 1$ to create a point $x$ in the Leech lattice $\Lambda_{24}$ with $||x||^2=8$. $ \frac{1}{2}...
0
votes
3answers
46 views

Is there a theorem or axiom which shows that permutations of step sequences through a lattice graph result in the same destination?

I have been searching for a theorem, lemma, or even an axiom which shows that the permutations of a step sequence in Taxicab Geometry result in the same destination in such a lattice graph. To ...
2
votes
0answers
37 views

$\Delta$ vs. $\delta$ in Lattice Theory

I'm learning about lattices and I'd like to confirm the difference between the density $\Delta$ and another density $\delta$. I would greatly appreciate if someone could confirm, correct, or even ...
1
vote
1answer
41 views

What does the notation for a group ${A_n}^*$ mean?

I was on a website that catalouges lattices. I was looking through the alternating group lattices and the dihedral lattices and there were two kinds for each. For example, there was $A_2$ and ${A_2}^*$...
1
vote
1answer
89 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 ...
3
votes
1answer
81 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 ...
0
votes
1answer
35 views

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 ...
2
votes
0answers
70 views

Continuity of a positive preserving operator between C(X) and C(Y)

I've been struggling with this question in Reed and Simon while I'm prepping for quals. Suppose that $T:C(X)\rightarrow C(Y)$ is a positive operator. Prove that T is continuous and $\Vert T\Vert_{...
0
votes
0answers
424 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 $\...
1
vote
0answers
44 views

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 ...