Annihilators of Banach Spaces

**Mimi89** · November 28th 2010, 09:54 AM

Hi All,

Could you please help with the following?

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

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

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

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

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

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

Now, I can show that T is well-defined and linear on the subspace $(ranf_{1}) x ... x (ranf_{n})=:K$ .
By the Hahn-Banach theorem, I can extend this to a function $G \in $ \mathbb{F}^{n} $'$ such that $G|_{K}=T$ and $\|G\|_{ ( \mathbb{F}^{n} )'} = \|T\|_{K'}$ . But where do I go from here?

**Opalg** · November 28th 2010, 12:13 PM

The key thing that you need to notice is that every linear functional $T$ on $\mathbb{F}^{n}$ is of the form $Ty = \alpha_1y_1 + \alpha_2y_2 + \ldots + \alpha_ny_n\;\,(y = (y_1,y_2,\ldots,y_n)\in\mathbb{F}^{n})$ , for some scalars $\alpha_1,\, \alpha_2,\,\ldots, \alpha_n$ . All you have to do then is to take $(y_1,y_2,\ldots,y_n) = (f_1(x),f_2(x),\ldots,f_n(x))$ .

**Mimi89** · November 29th 2010, 06:48 AM

Thanks a lot for this!
Could you please check I've got the last part, ( $( M_{o} ) ^{o} =M$ ), right?

So, want to show $M \subset ( M_{o} ) ^{o}\ , \ so \ take \ f \in M \leq X' \ \Rightarrow f \in X'$ Want to show $f \in ( M_{o} ) ^{o}$ ,

i.e. that (by defn of $( M_{o} ) ^{o}$ ) $f \in X' \ and \ f(x)=0 \ \forall x \in M_{o}$
i.e. (by defn of $M_{o}$ ) $f(x)=0 \ \forall x \in X : (f(x)=0 \forall f \in M )$
which is true, as we started with $f \in M$

Similarly for the other direction. What I don't get - where did we use the $Ker(f) \supset \bigcap_{i} Ker (f_{i})$ bit? And where in this last part did we use any of the previous bits of the question?

Thanks in advance!

**Opalg** · November 29th 2010, 08:18 AM

Originally Posted by Mimi89

What I don't get - where did we use the $Ker(f) \supset \bigcap_{i} Ker (f_{i})$ bit?

The place where you needed $\ker(f) \supset \bigcap_{i} \ker (f_{i})$ was right at the beginning of the question, in order to show that $T$ is well defined. If the definition $T: (f_{1}(x),...,f_{n}(x))\rightarrow f(x)$ is to make sense, it must be true that $f(x)$ depends only on $f_{1}(x),...,f_{n}(x)$ . So you need to be sure that if $(f_{1}(x),...,f_{n}(x)) = (f_{1}(y),...,f_{n}(y))$ then $f(x) = f(y)$ . But that is equivalent to saying that if $f_i(x-y) = 0$ for i=1,2,...,n, then $f(x-y)=0$ . That in turn is equivalent to the condition $\ker(f) \supset \bigcap_{i} \ker (f_{i})$ .

**Opalg** · November 29th 2010, 08:38 AM

Originally Posted by Mimi89

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

For the last part of the question, you are given a finite-dimensional subspace $M\subset X'$ . Let $f_1,\ldots,f_n$ be a basis for $M$ . Then

$M_o = \{ x \in X: f(x)=0\ \forall f \in M \} = \{x\in X:f_i(x)=0\ (1\leqslant i\leqslant n)\} = \bigcap_{i} \ker (f_{i}),$

and $(M_o)^o = \{f\in X':f(M_o)=0\} = \{f\in X':\ker(f)\supset \bigcap_{i} \ker (f_{i})\}.$

But it follows from the earlier part of the question that that last condition implies that $f$ is a linear combination of $f_1,\ldots,f_n$ , which says that $f\in M$ . Therefore $(M_o)^o\subset M$ . You have shown how to prove the reverse inclusion $M\subset (M_o)^o$ , and so $(M_o)^o = M$ .

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