Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [vector-lattices]

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

A Riesz space, lattice-ordered vector space or vector lattice is a vector space with a partial order which

  • is compatible with the structure of the vector space and
  • forms a lattice.

Many of spaces used in functional analysis carry a natural ordering such that they form a vector lattice.

Banach lattices are an important class of Riesz spaces.

155 questions
2
votes
1 answer

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?
cejvan
  • 222
2
votes
1 answer

Does Dedekind completeness imply the Archimeadean property?

Suppose that $A$ is a Riesz space (vector lattice). Recall the following terminology: A is Archimedean if for any $x,y\geq 0$ such that $n x\leq y$ for $n=1,2,\ldots$ is follows that $x=0$; A is Dedekind complete if every upper bounded subset of…
John Paul Littlefield
  • 29
1
vote
0 answers

About the infimum of two operators in vector lattices

A real vector space $E$ is said to be an ordered vector space whenever it is equipped with an order relation $\ge$ that is compatible with the algebraic structure of $E$. A Riesz space is an ordered vector space $E$ which for each pair of vectors…
I-love-math
  • 331
1
vote
1 answer

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?
seunglim
  • 71
0
votes
1 answer

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 ordered set such that (i) $x \leq y \Rightarrow x +…
roo
  • 5,598
0
votes
1 answer

If $A$ is a linear subspace of the Riesz space $E$, then $A=A^{dd}$.

If $A$ is a linear subspace of the Riesz space $E$, then $A=A^{dd}$. Definition. Let $E$ be a Riesz space. The elements $f$ and $g$ in $E$ are said to be disjoint, we write $f\perp g$, if $|f| \wedge |g|=0$. Definition. Let $E$ be a Riesz space and…
lap lapan
  • 2,188
0
votes
1 answer

Positive Basis of a Riesz Space

Does a Riesz space in general always has a positive basis? ae. if $E$ is a Riesz space, can we assume that there exists a set $B\subset E$ such that it is a basis of the vector space $E$ and for every $v\in B$, $v>0$? Proving for the special case…
Sagi Shadur
  • 571
0
votes
1 answer

The number of lattice points inside R-radius ball centered at origin

This is a question from Hoffstein cryptography book. I'm trying to show that $\lim_{R \to \infty}$$\frac{{\#(\mathbb{B}_R(\mathbf{0})}\cap L)}{{Vol(\mathbb{B}_R(\mathbf{0}))}}$$=\frac{1}{Vol(\mathcal{F})}$ for a lattice L and its fundamental domain…
Measurezero
  • 23