In this work a LaxMilgram
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 c^{2} = 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 H_{2}
supΦ_{2}(u,v)(Φ_{2}(v,v)^{1∕2})^{−1} exists and equals Φ_{2}(u,u)^{1∕2}. Let B : H_{1} ×H_{2} → K
be an orthocontinuous bilinear form satisfying:
 inf _{x≠0} sup_{y≠0}B(x,y)(Φ_{1}(x,x)^{1∕2}Φ_{2}(y,y)^{1∕2})^{−1} = γ exists and γ −δ is
in P^{+} for some δ in P^{+}.
 supB(x,y) exists and is in P^{+} for all y≠0 x ∈ H.
Then given any orthocontinuous linear functional ϕ on H_{2} whose kernel is
splitting there exists a unique element x_{0} in H_{1} such that ϕ(y) = B(x_{0},y) for all y in
H_{2}.
Moreover
δ^{−1} sup_{y≠0}ϕ(y)(Φ_{2}(y,y)^{1∕2})^{−1} −Φ_{1}(x_{0},x_{0})^{1∕2} ∈ P^{+} ∪{0}.
