Hi All,

Could you please help with the following?

Let X be a normed vector space with Banach dual space X'. Let $\displaystyle f_{1}, \ ... \ f_{n} $ be in X' such that $\displaystyle Ker(f) \supset \bigcap_{i} Ker (f_{i}) $. Show that

$\displaystyle T: (f_{1}(x),...,f_{n}(x))\rightarrow f(x) $

is a well-defined linear map on a subspace of $\displaystyle \mathbb{F}^{n}$. Extending T to all of $\displaystyle \mathbb{F}^{n}$, show that there are $\displaystyle \alpha_{1},...,\alpha_{n} \in \mathbb{F} $ such that $\displaystyle f=\sum_{j=1}^{n} \alpha_{j}f_{j} $

Conclude that, for any finite-dimensional subspace M of X', $\displaystyle ( M_{o} ) ^{o} =M $

Now, by definition, for $\displaystyle M \subset X, N \subset X' $

$\displaystyle M^{o}= \{f \in X': \ f(x)=0 \ \forall x \in M\}$

$\displaystyle N_{o}=\{ x \in X; \ f(x)=0 \ \forall f \in N \} $

Now, I can show that T is well-defined and linear on the subspace $\displaystyle (ranf_{1}) x ... x (ranf_{n})=:K $.

By the Hahn-Banach theorem, I can extend this to a function $\displaystyle G \in \( \mathbb{F}^{n} \)' $ such that $\displaystyle G|_{K}=T $ and $\displaystyle \|G\|_{ ( \mathbb{F}^{n} )'} = \|T\|_{K'} $. But where do I go from here?