Joined: Mon Nov 09, 2015 1:52 pm Posts: 426

We consider \(\displaystyle{\ell^2}\) with the usual inner product and \(\displaystyle{c_{0 0}\subseteq \ell^2}\) . We consider in \(\displaystyle{c_{0 0}}\) the inner product from \(\displaystyle{\ell^2}\)
and the norm generated by this inner product. Let
\(\displaystyle{M=\left\{x=\left(x_{n}\right)_{n\in\mathbb{N}}\in c_{00}: \sum_{n=1}^{\infty}\dfrac{x_{n}}{n}=0\right\}}\) .
Prove that there exists a unique \(\displaystyle{f\in c_{00}^{\star}=\mathbb{B}(c_{00},\mathbb{C})}\) such that \(\displaystyle{M=\rm{Ker}(f)}\) and deduce that
\(\displaystyle{M\subseteq c_{00}}\) is a closed linear subspace.

