Definition:Almost Convergent Sequence
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Definition
Let $\map {\ell^\infty} \R$ be the vector space of bounded sequences on $\R$.
Let $\sequence {x_n}_{n \mathop \in \N} \in \map {\ell^\infty} \R$.
We say that $\sequence {x_n}_{n \mathop \in \N}$ is almost convergent to $L$ if and only if:
- $\map \phi {\sequence {x_n}_{n \mathop \in \N} } = L$
for each Banach limit $\phi$.
Examples
Example: $\sequence {0, 1, 0, 1, \ldots}$ almost converges to $1/2$
Let $\sequence {x_n}_{n \in \N}$ be the sequence defined by:
- $x_n = \begin{cases} 0 & : n \equiv 0 \pmod 2 \\ 1 & : n \equiv 1 \pmod 2 \end{cases}$
where $\bmod$ denotes the congruence modulo.
Then $\sequence {x_n}_{n \in \N}$ almost converges to $1/2$.
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $20.5$: Generalised Banach Limits