An algebraic extension of the Lax-Milgram theorem

Gabriel Michael Miller Obi

Vol. 86 (1980), No. 2, 543–552

DOI: 10.2140/pjm.1980.86.543

Abstract

In this work a Lax-Milgram type theorem is proved for quadratic spaces over a division ring K with involution *, say, whose center contains an ordered domain P such that for every element a in K, aa∗ = |a|², (where |a|, the absolute value of a, is in P⁺ which is the set of positive elements of P), and for every element b in P⁺ there exists an element c in P⁺, denoted by b^1∕2, such that c² = b. Specifically, with the above assumptions on K, the following is proved:

Let (H_i,Φ_i) i = 1,2 be quadratic spaces over K such that for each u in H₂ sup|Φ₂(u,v)|(|Φ₂(v,v)|^1∕2)⁻¹ exists and equals |Φ₂(u,u)|^1∕2. Let B : H₁ ×H₂ → K be an orthocontinuous bilinear form satisfying:

1. inf _x≠0 sup_y≠0|B(x,y)|(|Φ₁(x,x)|^1∕2|Φ₂(y,y)|^1∕2)⁻¹ = γ exists and γ −δ is in P⁺ for some δ in P⁺.
2. sup|B(x,y)| exists and is in P⁺ for all y≠0 x ∈ H.

Then given any orthocontinuous linear functional ϕ on H₂ whose kernel is splitting there exists a unique element x₀ in H₁ such that ϕ(y) = B(x₀,y) for all y in H₂.

Moreover

δ⁻¹ sup_y≠0|ϕ(y)|(|Φ₂(y,y)|^1∕2)⁻¹ −|Φ₁(x₀,x₀)|^1∕2 ∈ P⁺ ∪{0}.

Mathematical Subject Classification 2000

Primary: 15A63

Secondary: 46C10

Milestones

Received: 8 September 1978

Revised: 1 May 1979

Published: 1 February 1980