Math 2.0

Functional analysis Definition

Ordered vector space

Builds on Vector spacePartial orderPositive cone

Definition

Let VV be a real vector space and let ≤\le be a partial order on VV. The pair (V,≤)(V, \le) is an ordered vector space if for all x,y,z∈Vx, y, z \in V and all real λ≥0\lambda \ge 0:

  1. if x≤yx \le y, then x+z≤y+zx + z \le y + z;
  2. if x≤yx \le y, then λx≤λy\lambda x \le \lambda y.

Remarks

Example: ℝn\R^n with the componentwise order is an ordered vector space. If xi≤yix_i \le y_i for all ii, then xi+zi≤yi+zix_i + z_i \le y_i + z_i and λxi≤λyi\lambda x_i \le \lambda y_i for λ≥0\lambda \ge 0. Non-example: on ℝ\R, let x⪯yx \preceq y when x=yx = y or (x,y)=(0,1)(x, y) = (0, 1). This is a partial order, but 0⪯10 \preceq 1 while 1⪯̸21 \not\preceq 2, so clause 1 fails. The set {x∈V:0≤x}\set{x \in V : 0 \le x} is a positive cone, and the order is recovered from it by x≤y  ⟺  y−x≥0x \le y \iff y - x \ge 0. So giving an ordered vector space is the same as giving a real vector space with a positive cone.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionMatrix, definition, no evidence levelMatrixdefinitionProposition, concept, no evidence levelPropositionconceptVector space, definition, no evidence levelVector spacedefinitionSet, definition, no evidence levelSetdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionPositive cone, definition, no evidence levelPositive conedefinitionOrdered vector space, definition, no evidence levelOrdered vector spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Matrix definition
  3. Proposition concept
  4. Vector space definition
  5. Set definition
  6. Square matrix definition
  7. Transpose definition
  8. Implication definition
  9. Predicate definition
  10. Ordered pair definition
  11. Conjugate transpose definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Hermitian matrix definition
  15. Subset definition
  16. Binary relation definition
  17. Positive semidefinite matrix definition
  18. Antisymmetric relation definition
  19. Reflexive relation definition
  20. Transitive relation definition
  21. Preorder definition
  22. Partial order definition
  23. Positive cone definition
  24. Ordered vector space definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordered_vector_space
About this page
TagX32V6Y5PYW
URIm2://X32V6Y5PYW
Kinddefinition
WritingW0 stub
MSC46A40, 06F20
DefinesOrdered vector space
UsesVector space, Partial order, Positive cone
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by