A Classification of Elements of Sequence Space $Seq(\mathbb{R})$

TL;DR

This paper introduces a new classification of the sequence space of real sequences based on asymptotic limit profiles, revealing a structured partition into seven distinct classes with explicit representatives and a complex connectivity graph.

Contribution

It provides a novel, constructive classification of $Seq( eal)$ using limit profiles, establishing a detailed macro- and micro-structure analysis with explicit sequences and connectivity relations.

Findings

Partition into seven disjoint classes based on limit profiles

Explicit representative sequences for each class

Complex connectivity graph with a global attractor

Abstract

The sequence space of all real-valued sequences, denoted $Seq(\mathbb{R})$, is typically investigated through the lens of infinite-dimensional vector spaces, utilizing Banach space norms or Schauder bases. This work proposes a complementary, constructive classification based instead on the asymptotic limit profile encoded by the pair $(\liminf a_n, \limsup a_n)$. We demonstrate that this perspective naturally partitions $Seq(\mathbb{R})$ into seven mutually disjoint macroscale blocks, covering behaviors from finite convergence to bounded and unbounded oscillation. For each block, we provide explicit closed-form representative sequences and establish that every constituent class possesses the cardinality of the continuum. Furthermore, we investigate the structural relationships between these blocks at two distinct levels of granularity. At the macroscale, we employ injective mappings to define an idealized connectivity graph, while at the microscale, we introduce a connection relation governed by the Hadamard (pointwise) product. This dual analysis reveals a rich directed graph structure where the block of finite convergent sequences functions both as the only block subspace and as a global attractor with no outgoing connections. Statistical comparisons between the idealized and realized adjacency matrices indicate that the pointwise product structure realizes approximately two-thirds of the theoretically possible macroscale relations. Ultimately, this partition-based framework endows the seemingly chaotic space $Seq(\mathbb{R})$ with a transparent, geometrically interpretable internal structure.