A Classification of Elements of Sequence Space Seq(ℝ)
- 1PhD, PStat, Biostatistics Division, Dalla Lana School of Public Health, University of Toronto,Toronto, ON, Canada
Abstract
The sequence space of all real-valued sequences, denoted Seq(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 an, limsup an). We demonstrate that this perspective naturally partitions Seq(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(ℝ) with a transparent, geometrically interpretable internal structure.
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BibTeX
@article{soltanifar2026classification,
title = {{A Classification of Elements of Sequence Space $Seq(\mathbbR)$}},
author = {Mohsen Soltanifar},
journal = {Mathematics Interdisciplinary Research},
year = {2026},
month = oct,
doi = {10.22052/mir.2026.259151.1595},
url = {https://doi.org/10.22052/mir.2026.259151.1595},
note = {CC BY},
}
RIS
TY - JOUR
TI - A Classification of Elements of Sequence Space Seq(ℝ)
AU - Mohsen Soltanifar
PY - 2026
DA - 2026/10/30
JO - Mathematics Interdisciplinary Research
DO - 10.22052/mir.2026.259151.1595
UR - https://doi.org/10.22052/mir.2026.259151.1595
AB - The sequence space of all real-valued sequences, denoted Seq(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(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(ℝ) with a transparent, geometrically interpretable internal structure.
SN - 2476-4965
ER -