Problem regarding the pettis integral

Assignment Help Basic Statistics

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

Questions Cloud

Finite dimensional banach space : 1. Show that in any finite dimensional Banach space (Rk with any norm), for any closed, convex set C and any point x not in C , there is at least one nearest point y in C ; in other words, ly - x l= infz∈C lz - x l.

Which of the following are measures of variability : Which of the following are measures of central tendency? Select all that apply

Because of such relationships : Let (X, l·l) be a real normed space, E a linear subspace, and h ∈ E t. Give a proof that h can be extended to be a member of X t (the Hahn-Banach the- orem, 6.1.4) based on Theorem 6.2.11. Hint: Let U := {x ∈ X : lx l 1}. (Because of such relation..

Determining the support planes of the cube : In R3, which planes through the origin are support planes of the unit cube [0, 1]3? Find all the support planes of the cube.

Problem regarding the pettis integral : 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.

How much energy is lost on the collision : A 220 gram ball of clay travelling east at 20.0 m/s collides perfectly In-elastically with a 300 gram ball of clay travelling 30.0o west of south at 10.0m/s. What are the speed and the direction of the resulting blob of clay?

Determining the converge to an element : Show that ( f dµ is well-defined and if ( 1 fn - g1 dµ → 0, then ( fn dµ converge to an element of S depending only on g, which will be called ( g dµ (Bochner integral).

Problem about the lipschitzian function : Let f be a Lipschitzian function on X and E a subset of X where f L on X is the same as for its restriction to E . Let x ∈ X \E . Does f on E uniquely determine f on X?

Set of continuous real-valued functions : The support of a real-valued function f on a topological space is the closure of {x : f (x ) /= 0}. Let L be the set of continuous real-valued functions on R with compact support.

User Account

All Pages