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Math Help - Folland:Real Analysis, section 5.2 #24

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    Folland:Real Analysis, section 5.2 #24

    suppose X is a Banach space, let X^\tilde and (X*)^\tilde be the natural images of X, X* in X**, X***, and let (X^\tilde)^0={F \in X***: F|_{X^\tilde}=0}. Then show that:

    (X*)^\tilde+(X^\tilde)^0=X***
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    Quote Originally Posted by frankmelody View Post
    suppose X is a Banach space, let X^\tilde and (X*)^\tilde be the natural images of X, X* in X**, X***, and let (X^\tilde)^0={F \in X***: F|_{X^\tilde}=0}. Then show that:

    (X*)^\tilde+(X^\tilde)^0=X***
    This becomes easier if you use a better notation. Write J:X\to X^{**} and K:X^*\to X^{***} for the canonical embeddings. Let J^*:X^{***}\to X^* be the adjoint of J, so that (J^*\Phi)(x) = \Phi(Jx)\ \ (\Phi\in X^{***},\,x\in X).

    For \Phi\in X^{***}, let \Psi = K(J^*\Phi). Then it's just a matter of definition-chasing to check that \Phi - \Psi \in (JX)^0. That shows that X^{***} = KX^* + (JX)^0. (In fact, the sum is a direct sum.)
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