# Functional not attaining its norm

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 $$||x||_{\ell^1}+||x||_{\ell^p}=1$$?

• Your space is isomorphic to $\ell^1$, hence non-reflexive, so James' Theorem guarantees the existence. Exhibiting one if of course a different problem.
– Ruy
Nov 28 '20 at 14:28
• Yes, indeed thanks to James' Theorem I know I'm looking for something which exists, but I would like to exhibit one. Nov 28 '20 at 15:52
• Try pairing a vector in your space with the $\ell^\infty$ sequence $(1-1/n)_n$.
– Ruy
Nov 28 '20 at 17:01
• 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! 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 $$k\rightarrow\infty$$, $$\|F\|^*= 1$$.

• 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 $1/2$.
– Ruy
Nov 28 '20 at 23:45