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**Mimi89** In the Banach space $\displaystyle l^{1} $ the sequence $\displaystyle (x_{k})$ of elements is defined by

$\displaystyle x_{k} = e_{k} -2e_{k+1} + e_{k+2} $

where $\displaystyle e_{k} $ is the co-ordinate sequence $\displaystyle ( \delta_{kj} ) $. Let M be the closed linear span of the elements $\displaystyle x_{1}, x_{2}, ... $. By considering the liner functional f on $\displaystyle l^{1} $ defined by

$\displaystyle f((\alpha _{j} )) = \sum_{j=1}^{\infty} \alpha _{j} $

or otherwise, prove that M does not coincide with $\displaystyle l^{1} $.