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 Post subject: Closed linear subspacePosted: Sat Mar 05, 2016 6:39 pm
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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.

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