Simultaneous Linear Inequalities:
Yesterday and Today

1  Agenda

Linear inequality implies linearity and order. When combined, the two produce an ordered vector space. Each linear inequality in the simplest environment of the sort is some half-space. Simultaneity implies many instances and so evolves intersections of half-spaces. This yields polyhedra as well as arbitrary convex sets, identifying the theory of linear inequalities with convexity.

Convexity stems from the remote ages and reigns in the federation of geometry, optimization, and functional analysis. Convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability.

This talk addresses the origin and the state of the art of the relevant areas with a particular emphasis on the Farkas Lemma. Our aim is to demonstrate how Boolean valued analysis may be applied to simultaneous linear inequalities with operators. This particular theme is another illustration of the deep and powerful technique of “stratified validity” which is characteristic of Boolean valued analysis.

2  Environment

Assume that X is a real vector space, Y is a Kantorovich space also known as a complete vector lattice or a Dedekind complete Riesz space. Let B:= B(Y) be the base of Y, i.e., the complete Boolean algebras of positive projections in Y; and let m(Y) be the universal completion of Y. Denote by L(X,Y) the space of linear operators from X to Y. In case X is furnished with some Y-seminorm on X, by L^(m)(X,Y) we mean the space of dominated operators from X to Y. As usual, {T ≤ 0}:={x ∈X : Tx ≤ 0}; ker(T)=T^-1(0) for T ∈ L(X,Y).

3  Kantorovich's Theorem

If W is ordered by W₊ and A(X) – W₊= W₊ – A(X)=W, then

(∃𝔛≥ 0) 𝔛A=B ↔ {A ≤ 0}⊂ {B ≤ 0}.

4  The Alternative

 Let X be a Y-seminormed real vector space, with Y a Kantorovich space. Assume that A₁,...,A_N and B belong to  L^(m)(X,Y).

Then one and only one of the following holds:

(1) There are x ∈ X and b, b' ∈ B such that b'≤ b and

b'Bx > 0, bA1 x ≤ 0,..., bAN x ≤ 0.

(2) There are α₁,..., α_N ∈ Orth(m(Y))₊ such that B=∑_k=1^Nα_k A_k.

5  Inhomogeneous Inequalities

Let X be a Y-seminormed real vector space, with Y a Kantorovich space. Assume given some dominated operators A₁,...,A_N, B ∈ L^(m)(X,Y) and elements u₁,..., u_N,v ∈ Y. The following are equivalent:

(1) For all b∈ B the inhomogeneous operator inequality bBx ≤ bv is a consequence of the consistent simultaneous inhomogeneous operator inequalities bA₁ x ≤ bu₁,..., bA_N x ≤bu_N, i.e.,

{bB ≤ bv} ⊃ {bA1 ≤ bu1}∩... ∩{bAN ≤ buN}.

(2) There are positive orthomorphisms α₁,..., α_N ∈ Orth(m(Y)) satisfying

B= N ∑ k=1  αk Ak;    v ≥ N ∑ k=1  αk uk.

6  Freedom and Inequality

Abstraction is the freedom of generalization. Freedom is the loftiest ideal and idea of man, but it is demanding, limited, and vexing. So is abstraction. So are its instances in convexity, hence, in simultaneous inequalities.

The freedom of set theory empowered us with the Boolean-valued models yielding a lot of surprising and unforeseen visualizations of the ingredients of mathematics. Many promising opportunities are open to modeling the powerful habits of reasoning and verification. Convexity, the theory of simultaneous linear inequalities in disguise, is a topical illustration of the wisdom and strength of mathematics, the ever fresh art and science of calculus.

Freedom presumes liberty and equality. Inequality paves way to freedom.