Vol. 86, No. 2, 1980

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An algebraic extension of the Lax-Milgram theorem

Gabriel Michael Miller Obi

Vol. 86 (1980), No. 2, 543–552

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 b12, such that c2 = b. Specifically, with the above assumptions on K, the following is proved:

Let (Hi,Φi) i = 1,2 be quadratic spaces over K such that for each u in H2 sup|Φ2(u,v)|(|Φ2(v,v)|12)1 exists and equals |Φ2(u,u)|12. Let B : H1 ×H2 K be an orthocontinuous bilinear form satisfying:

  1. inf x0 supy0|B(x,y)|(|Φ1(x,x)|12|Φ2(y,y)|12)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 y0 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.


δ1 supy0|ϕ(y)|(|Φ2(y,y)|12)1 −|Φ1(x0,x0)|12 P+ ∪{0}.

Mathematical Subject Classification 2000
Primary: 15A63
Secondary: 46C10
Received: 8 September 1978
Revised: 1 May 1979
Published: 1 February 1980
Gabriel Michael Miller Obi