Mathematics Interdisciplinary Research

Mathematics Interdisciplinary Research

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

Document Type : Original Scientific Paper

Author
Mohsen Soltanifar
‎PhD‎, ‎PStat,‎ Biostatistics Division,‎ Dalla Lana School of Public Health, University of Toronto,Toronto‎, ‎ON‎, ‎Canada
10.22052/mir.2026.259151.1595
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 as both the only 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.
Keywords
Convergent sequences
Divergent sequences
Sequence spaces
Limits
Digraphs
Subjects
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Mathematics Interdisciplinary Research
Volume 11, Issue 4
Autumn 2026
Pages 407-434