Problem regarding the pettis integral

Reference no: EM13990817

1. (Continuation.) If g is a function from X into S, then t is called the Pettis integral of g if for all u ∈ St, ( u(g) dµ = u(t ). Show that the Bochner integral of g, when it exists, as defined in Problem 1, is also the Pettis integral.

2. Prove that every Hilbert space H over the complex field C is reflexive. Hints: Let C (h)( f ) := ( f, h) for f, h ∈ H . Then C takes H onto H t by Theorem 5.5.1. Show using (5.3.3) that C (h) t = h for all h ∈ H . Let (C ( f ), C (h))t := (h, f ) for all f, h ∈ H . Show that this defines an inner product (·,·)t on H t × H t such that [(ψ, ψ )t]1/2 = ψ t for all ψ ∈ H t; and H t is then a Hilbert space. Apply Theorem 5.5.1 to H t and (H t)t to conclude that I tt is onto (H t)t.

Reference no: EM13990817

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