4
$\begingroup$

Let $f:(c_{00},\|\cdot\|_1)\to \mathbb C $ be a non zero continuous linear functional. The number of Hahn-Banach extensions of f to $(\ell^1,\|\cdot\|_1)$ is

  1. one
  2. two
  3. infinite
  4. three

I have no idea how to get the result.

Hahn Banach theorem on normed linear space: If I have a linear functional defined on some subspace then I can extend it on the whole vector Space with same norms.

Here $\|x\|_1= \sum_{n=1}^\infty |x_n|$,where $x=(x_1,x_2....)\in c_{00} = $ the set of all sequences whose finite terms are non-zero.

If $x\in \ell^1$, then $\|x\|_1= \sum_{n=1}^\infty |x_n|<\infty $.

Someone help. Thanks.

$\endgroup$
3
$\begingroup$

Let my give some hints which should help to solve the problem:

  • If we have two different extensions, all convex combinations are still extensions, hence, the answer is either $1$ or $\infty$.
  • $c_{00}$ is dense in $\ell^1$. And we get only unique extension if the subspace is dense in X
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.