High-Dimensional Sequence Spaces

• High-dimensional sequence spaces are versatile constructs defined by large or infinite index sets, revealing unique geometric and probabilistic properties.
• They underpin modern analysis and coding theory by providing frameworks for studying measure concentration, operator compactness, and quantitative entropy.
• These spaces support advanced methodologies in invariant subspace construction, duality analysis, and the design of effective coding systems.

High-dimensional sequence spaces comprise a spectrum of functional analytic and combinatorial structures where the dimension or index set of sequences is large—finite but massive, or genuinely infinite—yielding distinctive geometric, probabilistic, and structural properties. Such spaces form the backbone of modern analysis, probability, coding theory, and mathematical biology, where large systems naturally arise. Their theory covers topics from universality phenomena, spaceability, entropy geometry, and operator compactness to applications in information theory and the structure of evolutionary optimization.

1. Foundations and Notable Constructions

Classical sequence spaces, such as $\ell_p(X)$ and $c_0(X)$ for a Banach space $X$, represent prototypical examples, with

$$\ell_q(X) = \left\{\, x=(x_n) \in X^{\mathbb{N}} : \|x\|_{\ell_q(X)} = \left( \sum_{n=1}^\infty \|x_n\|_X^q \right)^{1/q} < \infty \right\},\quad (0<q<\infty)$$

and

$$c_0(X) = \left\{ x=(x_n) : \lim_{n\to\infty} \|x_n\|_X = 0 \right\}.$$

Generalizations include vector-valued, weighted, Lorentz, Orlicz, and difference sequence spaces. For example, the difference-mean sequence space $\ell(r,s,t,p;\Delta^{(m)})$ encodes generalized sums with difference operators and a sequence of exponents, admitting a complete paranormed structure and explicit Schauder bases (Maji et al., 2013). Similarly, generalised Morrey sequence spaces $m_{\varphi,p}(\mathbb{Z}^d)$ control the local $p$-summability of a sequence over dyadic cubes weighted by a function $\varphi$ of the scale, interpolating between $\ell^p$ and $\ell^\infty$ in high dimensions (Haroske et al., 19 Feb 2025). More exotic constructions appear in finite geometry, such as the parameterized families of scattered subspaces and scattered sequences over finite fields related to maximum rank-distance codes (Bartoli et al., 2024).

2. Geometry, Concentration, and Universality Phenomena

A hallmark of high-dimensional sequence spaces is the emergence of geometric phenomena absent in low dimensions. The “concentration of measure” states that Lipschitz functions on high-dimensional spheres or simplexes become almost constant, with fluctuations of order $O(1/\sqrt{n})$ for ambient dimension $n$. For quasi-species models, mapping the frequency simplex (probability distributions over $n$ types) to the positive orthant of $S^{n-1}$ via $x_i \mapsto y_i = \sqrt{x_i}$ yields that the mean fitness function $f(Y) = \sum a_i y_i^2$ is Lipschitz with constant $L=2a_{\max}$; Lévy’s Lemma then enforces that, for large $n$,

$$\Pr\big\{ |f(Y) - M_f| > \epsilon \big\} \leq 2\exp\left( -\frac{C n \epsilon^2}{4 a_{\max}^2} \right),$$

so almost all high-dimensional populations exhibit nearly universal mean fitness, independent of mutation and initialization (Madhok, 2016). This mechanism applies to any Lipschitz observable on the sequence space, explaining the robust “self-averaging” behavior of large evolutionary or statistical systems.

3. Structure and Large Subspaces: Spaceability and Invariant Sequence Spaces

High-dimensional sequence spaces frequently retain vast linearly structured subsets even after substantial removal of “small” or “tame” subspaces. If $E$ is an invariant sequence space over $X$ (satisfying zero-free invariance and coordinate-wise domination), then for any countable family of subspaces $\{\ell_q(X)\}_{q\in\Gamma}$, the complement $E \setminus \bigcup_{q\in\Gamma} \ell_q(X)$, if non-empty, contains a closed infinite-dimensional subspace—this is termed spaceability (Botelho et al., 2010). The argument involves block-diagonal constructions exploiting the infinite dimensionality and coordinatewise control inherent in $E$. This phenomenon persists in Banach and quasi-Banach cases, generalizing to Orlicz, Lorentz, and vector-valued settings, and is insensitive to the removal of spaces such as $c_0(X)$. Thus, the “thin” complements of classic summability or vanishing-at-infinity ideals inside $\ell_p$, Orlicz, or generalized sequence spaces remain “large” in the algebraic-geometric sense.

4. Quantitative Geometry: Entropy Numbers and Embedding Complexity

The geometry of high-dimensional sequence spaces is quantitatively encoded by entropy numbers of operators acting between finite or infinite product sequences spaces. Given linear maps between vector-valued $\ell_p$ and $\ell_q$ spaces, two-term estimates of the entropy numbers capture a “phase transition” between effective rank and logarithmic regimes, controlled by the parameters $p$, $q$, the building-block operator, and the ambient dimension $m$:

$$e_n(T^{(m)}) \simeq \max \left\{ \sup_{1\leq k\leq n} \left( \frac{k}{n} \right)^{1/p - 1/q} e_k(T_0),\; \|T_0\|\left( \frac{\log(m/n+1)}{n} \right)^{1/p - 1/q} \right\}$$

(Edmunds et al., 2013). Such estimates describe the covering behavior (“widths”) of the high-dimensional balls, reflect the “curse of dimensionality,” and are critical in approximation theory, the study of function-space embeddings (via wavelet decompositions), and learning theory, where they bound the code length or sample complexity required in high-dimensional models.

5. Operator Theory, Compactness, and Duality

Operator theory in high-dimensional sequence spaces reveals intricate connections between sequence structure and compactness. In the generalized Morrey sequence spaces, the natural embeddings

$$\ell^{\infty}(\mathbb{Z}^d) \hookrightarrow m_{\varphi,p}(\mathbb{Z}^d) \hookrightarrow \ell^p(\mathbb{Z}^d)$$

are continuous under concrete conditions on $\varphi$, but are never compact for infinite index sets, and strict singularity can manifest, generalizing Kato’s phenomenon (Haroske et al., 19 Feb 2025). Paranormed difference sequence spaces $\ell(r,s,t,p;\Delta^{(m)})$ admit explicit dual characterizations ($\alpha$, $\beta$, $\gamma$-duals) and operator-norm estimates for matrices acting into classical sequence spaces. Measures of noncompactness (Hausdorff-type) can be described asymptotically via the $\ell_q$-decay of “far” rows of the operator matrix, supporting a fine-grained control of compact and non-compact behavior in high-dimensions (Maji et al., 2013).

6. Discrete Geometry and Finite Field Scenarios

High-dimensional sequence spaces over finite fields play a critical role in combinatorial geometry and coding theory. Scattered subspaces, constructed via $q$-linearized polynomials, generate exceptional Maximum Rank-Distance (MRD) codes of large dimension and minimality. Recent results have classified infinite families of indecomposable, exceptional scattered sequences for arbitrary order $m$ ($m\geq3$), producing genuine building blocks for rank-metric codes and novel linear sets in high-dimensional projective geometry. The equivalence theory and combinatorics ensure a rapidly growing number of inequivalent families, linked geometrically to maximal intersection properties and combinatorially to code minimality (Bartoli et al., 2024).

7. Non-classical Sequence Spaces and High-Dimensional Phenomena

Beyond classical settings, new families such as the generalized Morrey sequence spaces $m_{\varphi,p}(\mathbb{Z}^d)$ formalize local $L^p$-control across scales. Their high-dimensional complexity is determined by scalings like $2^{j d (1/p_2-1/p_1)}$ in embedding criteria—highlighting the volume explosion in large $d$ (Haroske et al., 19 Feb 2025). Difference sequence spaces perturbed by generalized means demonstrate how one can design new high-dimensional, complete, and structurally rich spaces, with controlled Schauder bases, duality, and operator behavior (Maji et al., 2013). These generalizations admit a flexible analytic toolkit for studying discretized partial differential equations, data encodings, and non-standard summability phenomena in high dimensions.

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High-dimensional sequence spaces thus form an extensive and interconnected area at the confluence of geometry, analysis, probability, and combinatorics. Their study leverages the architecture of infinite-dimensional topology, sharp quantitative geometry, duality, and measure concentration, revealing broad universality, large hidden structures, and robust asymptotic behaviors that govern the analysis, modeling, and optimization of large and complex systems (Madhok, 2016, Botelho et al., 2010, Edmunds et al., 2013, Maji et al., 2013, Haroske et al., 19 Feb 2025, Bartoli et al., 2024).