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Cross-posted from Math SE.

The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:

Does there exist an "explicitly definable" generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ that is stronger than the linear and stable closure of the Cauchy limit?

I'd be curious to know what such a $\operatorname{Lim}$ might be.

As Gerald pointed out, the concept of an almost convergent sequence is relevant. However, I don't have the positivity property of Banach limits since $\operatorname{Lim}(n \mapsto a^n) = (1-a)^{-1}$ which is negative for $a > 1$.

Cross-posted from Math SE.

The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:

Does there exist an "explicitly definable" generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ that is stronger than the linear and stable closure of the Cauchy limit?

I'd be curious to know what such a $\operatorname{Lim}$ might be.

As Gerald pointed out, the concept of an almost convergent sequence is relevant.

Cross-posted from Math SE.

The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:

Does there exist an "explicitly definable" generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ that is stronger than the linear and stable closure of the Cauchy limit?

I'd be curious to know what such a $\operatorname{Lim}$ might be.

Cross-posted from Math SE.

The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:

Does there exist an "explicitly definable" generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ that is stronger than the linear and stable closure of the Cauchy limit?

I'd be curious to know what such a $\operatorname{Lim}$ might be.

As Gerald pointed out, the concept of an almost convergent sequence is relevant. However, I don't have the positivity property of Banach limits since $\operatorname{Lim}(n \mapsto a^n) = (1-a)^{-1}$ which is negative for $a > 1$.

Cross-posted from Math SE.

The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. One question I asked is whether there exists a generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ such that

1. $\operatorname{Lim}$ is stronger than the stable and linear closure of the Cauchy limit.

2. $\operatorname{Lim}$ can be explicitly defined in terms of the Cauchy limit, e.g. $\operatorname{Lim} = \lim \circ f$ where $f : X^\mathbb{N} \rightarrow X^\mathbb{N}$ is some sequence transform.

I'd be curious to know if anyone knows what such a $\operatorname{Lim}$ might be (if it exists). Even an explicit characterization of a generalized limit satisfying 1 alone would be interesting.

Cross-posted from Math SE.

The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:

Does there exist an "explicitly definable" generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ that is stronger than the linear and stable closure of the Cauchy limit?

I'd be curious to know what such a $\operatorname{Lim}$ might be.