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

TL;DR

This paper introduces a finite, constructive partition of $Seq(\mathbb{R})$ by the limit profile $(\liminf a_n,\limsup a_n)$, yielding seven blocks each with continuum cardinality and explicit representative sequences. It develops a two-tier connectivity analysis: macroscale via injective block-to-block mappings giving a global adjacency pattern, and microscale via Hadamard-product connectors that realize a subset of these relations. The block of convergent sequences $G$ emerges as a linear subspace and global attractor, while the macroscale graph contains 42 potential relations and the microscale realization covers about two-thirds of these, yielding a dense yet non-exhaustive connectivity framework. Collectively, the partition provides a geometrically interpretable internal structure to $Seq(\mathbb{R})$ that complements traditional Banach-space classifications and suggests avenues for extending to other products and finer asymptotic descriptors.

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.

Figures (3)

• Figure 1: Partition of $Seq(\mathbb{R})$ by the $(L_1,L_2)$-plane, where $L_1(a)=\liminf a_n$ and $L_2(a)=\limsup a_n$, into seven admissible regions labelled $A,\dots,G$ satisfying $L_1 \le L_2$. Each region corresponds to a distinct asymptotic behaviour class of real-valued sequences (finite convergent, finite–finite divergent, and the various one-sided or two-sided infinite cases).
• Figure 2: Heptagon with with macroscale directed connectivity graph between seven blocks (vertices) $A,B,C,D,E,F,G$ and directed edges from $A,\dots,F$ to $G$. Each edge is tagged by the global map $T_G$, while the edge $F\to G$ also represents the specific map $T_{(F,G)}$.
• Figure 3: Heptagon with microscale directed connectivity graph between the seven blocks $A,B,C,D,E,F,G$ of the sequence space $Seq(\mathbb{R})$.

Theorems & Definitions (28)

• Definition 2.1: Sequence space $Seq(\mathbb{R})$
• Definition 2.2: Limit profile of the sequence
• Proposition 2.3: General properties of $\liminf$ and $\limsup$
• Corollary 2.4
• Theorem 2.5: Relationship between $\liminf$, $\limsup$, and the usual limit
• Remark 2.6
• Definition 2.7: Vector space structure on $Seq(\mathbb{R})$
• Definition 2.8: Hamel basis
• Remark 2.9: Cardinality of a Hamel basis of $Seq(\mathbb{R})$
• Definition 2.10: Schauder basis