In mathematical analysis, a Banach limit is a continuous linear functional ϕ : ℓ ∞ → C {\displaystyle \phi :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space ℓ ∞ {\displaystyle \ell ^{\infty }} of all bounded complex-valued sequences such that for all sequences x = {\displaystyle x=}, y = {\displaystyle y=} in ℓ ∞ {\displaystyle \ell ^{\infty }}, and complex numbers α {\displaystyle \alpha } : ϕ = α ϕ + ϕ {\displaystyle \phi =\alpha \phi +\phi } ; if x n ≥ 0 {\displaystyle x_{n}\geq 0} for all n ∈ N {\displaystyle n\in \mathbb {N} }, then ϕ ≥ 0 {\displaystyle \phi \geq 0} ; ϕ = ϕ {\displaystyle \phi =\phi }, where S {\displaystyle S} is the shift operator defined by n = x n + 1 {\displaystyle _{n}=x_{n+1}} ; if x {\displaystyle x} is a convergent sequence, then ϕ = lim x {\displaystyle \phi =\lim x}. Wikipedia