Banach limit

In mathematical analysis, a Banach limit is a continuous linear functional ${\displaystyle \phi :\ell ^{\infty }\to \mathbb {C} }$ defined on the Banach space ${\displaystyle \ell ^{\infty }}$ of all bounded complex-valued sequences such that for all sequences ${\displaystyle x=(x_{n})}$, ${\displaystyle y=(y_{n})}$ in ${\displaystyle \ell ^{\infty }}$, and complex numbers ${\displaystyle \alpha }$:

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