Definition:Almost Convergent Sequence

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