Consider the space $(\ell^1,||\cdot||_{\ell^1}+||\cdot||_{\ell^p})$ for some $p\in (1,\infty)$. Can someone provide an example of any $F:\ell^1\to \mathbb{R}$ linear and continuous such that $F(x)\neq ||F||^*$ for any $x\in \ell^1$ with $||x||_{\ell^1}+||x||_{\ell^p}=1$?

  • $\begingroup$ Your space is isomorphic to $\ell^1$, hence non-reflexive, so James' Theorem guarantees the existence. Exhibiting one if of course a different problem. $\endgroup$
    – Ruy
    Nov 28 '20 at 14:28
  • $\begingroup$ Yes, indeed thanks to James' Theorem I know I'm looking for something which exists, but I would like to exhibit one. $\endgroup$ Nov 28 '20 at 15:52
  • $\begingroup$ Try pairing a vector in your space with the $\ell^\infty$ sequence $(1-1/n)_n$. $\endgroup$
    – Ruy
    Nov 28 '20 at 17:01
  • $\begingroup$ Yes, I tried with that functional. It works if I consider the norm $||\cdot||_{\ell^1}$, but with the norm I wrote above I have not been able to prove that the functional does not attain the norm. Anyway, I'll keep trying thank you! $\endgroup$ Nov 28 '20 at 17:09

Let $\displaystyle{F(x)=\sum_{n=1}^\infty x_n}$. $|F(x)|\leq \|x\|_1<\|x\|_1+\|x\|_p$ so that $\|F\|^*\leq 1$. Let $u_k$ be defined by $u_{k,n}=1/k$ if $n\leq k$ and $u_{k,n}=0$ if $n> k$. Since $F(u_k)=1$ and $\|u_k\|_1+\|u_k\|_p= 1+k^{1/p-1}\rightarrow 1$ as $k\rightarrow\infty$, $\|F\|^*= 1$.

  • $\begingroup$ Nice! I wonder how to compute the norm of other functionals in this space. It will not always be the same as the $\ell^\infty$ norm, since the projection on a coordinate has norm $1/2$. $\endgroup$
    – Ruy
    Nov 28 '20 at 23:45

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