 To derive this from the M. Riesz extension theorem, define a convex cone "K" ⊂ R×"V" by

Define a functional "φ"1 on R×"U" by

One can see that "φ"1 is "K"-positive, and that "K" + (R × "U") = R × "V".Therefore "φ"1 can be extended to a "K"-positive functional "ψ"1 on R×"V".Then

is the desired extension of "φ".