A simple proof of Singer’s representation theorem

Abstract

Let Ω \Omega be a compact Hausdorff space and X X a Banach space. Singer’s theorem states that under the dual pairing ( f , m ) ↦ ∫ ⟨ f , d m ⟩ (f,m)\mapsto \int \langle f,dm\rangle , the dual space of C ( Ω ; X ) C(\Omega ;X) is isometric to r c a b v ( Ω ; X ′ ) rcabv (\Omega ;X’) . Using the Hahn-Banach theorem and the (scalar) Riesz representation theorem, a proof of Singer’s theorem is given which appears to be simpler than the proofs supplied earlier by Singer (1957, 1959) and Dinculeanu (1959, 1967).