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:
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 and for the canonical embeddings. Let be the adjoint of , so that .
For , let . Then it's just a matter of definition-chasing to check that . That shows that . (In fact, the sum is a direct sum.)