the problem may be posted again!
Let X be a normed vector space with Banach dual space X ′,let f be a non-zero element of X′, and let x0 be an element of X \ ker(f ). Show that every element x of X may be written in the form x = αx0 + y, with α in C and y in ker(f ). Deduce that ({f }◦down)◦up coincides with the subspace Cf .
Prove that, for any finite-dimensional subspace M of X′, (M◦down)◦up = M.
[M◦up = {f ∈ X′: f (x) = 0∀x ∈ M}, M◦down= {x ∈ X : f (x) = 0∀f ∈ M }].
Does anyone know how to do this question?
Let be a basis for M. Define a linear map (where or is the scalar field) by . Then . So induces an isomorphism from to . Write for the quotient map.
If then vanishes on and therefore gives a well-defined linear functional on the quotient space, given by . Define by . Notice that (the i'th coordinate of z), for .
Every linear functional on is a linear combination of coordinate functionals, so there exist such that . But this means that for all . Thus .
That proves that . The reverse inclusion is obvious.