Partial Differential Equations and Inverse Problems in Non-Separable Banach Spaces

Introduction

The theory of partial differential equations has historically evolved in close interaction with functional analysis, particularly within the framework of separable Banach and Hilbert spaces. Sobolev spaces, compact embeddings, Galerkin schemes, and spectral decompositions constitute the backbone of modern PDE analysis and its applications [1-5]. Nevertheless, these tools rely—often implicitly—on separability assumptions that fail in a variety of mathematically and physically relevant settings.

NSBS arise naturally in numerous contexts: spaces of bounded functions indexed by uncountable sets, spaces of essentially bounded observables in control theory, configuration spaces of large or infinite-agent systems, and dual spaces endowed with weak* topologies [6,7]. Despite their ubiquity, NSBS remain underrepresented in the mainstream PDE literature, primarily due to the loss of compactness, metrisability, and countable approximations.

The present article addresses this gap by developing a coherent theory of partial differential equations and inverse problems formulated directly on NSBS, without recourse to separable reductions. The guiding principle is methodological rather than merely technical: instead of forcing non-separable problems into separable frameworks, we adapt the analytic tools themselves to the intrinsic structure of NSBS (Da Prato & Zabczyk, 1992; Engel & Nagel, 2000; Kato, 1995; Nashed, 1976) [2,8-10].

Motivation and context

Several contemporary problems motivate this investigation:

a. In nonlinear dynamics and continuum models with infinitely many degrees of freedom, state spaces often exceed separable descriptions.

b. In inverse problems, the space of admissible parameters or controls may be inherently non-separable, particularly in identification problems with weak observational constraints [11,12].

c. In applied fields such as control theory, economics, and data-driven modelling, bounded but non-separable function spaces frequently arise as natural domains.

In such situations, classical existence and uniqueness results either fail or become inapplicable due to their reliance on compactness arguments or separable bases.

Objectives and contributions

The main objectives of this article are fourfold:

 i. To formulate a general framework for PDEs on non-separable Banach spaces, emphasising operator-theoretic and weak-topological methods.

ii. To construct explicit examples of PDEs posed on EBNS that illustrate both continuity with, and divergence from, the classical theory.

iii. To develop a functional-analytic approach to inverse problems in nonlinear dynamics, tailored to non-separable environments. iv. To establish adapted existence and uniqueness results, identifying the minimal structural assumptions required in the absence of separability.

Rather than presenting isolated results, the article aims to provide a unified conceptual framework consistent with a broader programme of analysis on EBNS.

Functional-Analytic Preliminaries on Non-Separable Banach Spaces

The transition from separable to non-separable Banach spaces involves a profound structural change in the topology and geometry of the ambient functional setting. In the separable case, sequential arguments, compact embeddings, and dense countable sets play a decisive role in existence and approximation theorems. By contrast, non-separable Banach spaces lack countable bases, are frequently non-metrisable in their weak or weak* topologies, and require a reformulation of compactness, duality, and convergence notions. This section recalls the essential concepts and establishes the analytic tools that will later support the formulation of partial differential equations and inverse problems on EBNS (Megginson, 1998; Rudin, 1991; Schaefer & Wolff, 1999) [5-7].

Non-Separability and Weak-Topological Framework

This generalised convergence underpins the analysis of PDEs in EBNS, where approximating sequences may not exist.

Approximate Compactness

A central obstacle in NSBS analysis is the failure of the Rellich–Kondrachov compact embedding and other compactness principles [6]. To overcome this, we employ the notion of approximate compactness:

Weak and Weak Operator Topologies*

Duality and Bi dual Extensions

Approximate Operators and Perturbations

In non-separable settings, compact operators form a negligible subset of L(X), often failing to capture essential analytical behaviour [2]. We therefore employ the class of approximately compact operators, defined as follows:

Definition 2.5.1 (Approximately Compact Operator). An operator K∈L(X) is approximately compact if it maps bounded subsets of X into approximately compact subsets of X.

Such operators preserve the essential features of compactness without requiring separability. They provide the natural perturbative framework for studying PDEs and inverse problems in EBNS, extending classical results on compact perturbations of Fredholm operators.

Canonical Non-Separable Examples

The following spaces illustrate the main structural phenomena encountered in the forthcoming sections:

i. l∞(Γ): the Banach space of bounded scalar functions on an uncountable index set Γ, endowed with the supremum norm. This space is non-separable, with (l∞ (Γ))\* = ba(Γ), the space of finitely additive measures.

ii. C(K): the space of continuous scalar functions on a compact Hausdorff space K that is non-metrisable. Its dual is the space of regular Borel measures M(K), which may itself be non-separable.

iii. Dual spaces of separable Banach spaces, such as L∞ (Ω) = (L1 (Ω))\* when Ωcarries a nonatomic measure, are typically non-separable in the norm topology but admit weak* compact unit balls by Banach–Alaoglu. These examples will serve as concrete domains for the PDE models and inverse problems in sections 4 and 5.

Summary and Analytical Outlook

The absence of separability compels a methodological shift from sequence-based to net-based arguments, from compactness to approximate compactness, and from countable decompositions to measure-theoretic and weak* constructions. The functional-analytic notions introduced in this section provide the essential vocabulary for the forthcoming development:

a. Section 3 will define abstract differential operators and PDEs on EBNS using the weak and weak* frameworks.

c. Section 4 will illustrate the general constructions in l∞ (Γ) and C(K).

b. Sections 6–7 will employ approximate compactness to establish existence and uniqueness theorems adapted to non-separable environments.

Abstract Formulation of PDEs in NSBS

This section develops an abstract framework for PDEs posed on NSBS, with particular emphasis on elliptic, parabolic, and evolution-type equations. The guiding principle is that PDEs are formulated and analysed as operator equations on NSBS, using weak and weak* topologies, approximate compactness, and bi dual extensions, rather than separability-based compactness or spectral decompositions (Da Prato & Zabczyk, 1992; Kato, 1995; Rudin, 1991; Showalter, 1997; Yosida, 1980) [2,5,8,14,15].

Operator-Theoretic Formulation of PDEs

Let X and Y be Banach spaces, possibly non-separable. A (linear or nonlinear) partial differential equation is modelled abstractly as A(u) = f, where A:D(A) ⊂ XY is a (possibly unbounded) operator encoding the differential structure of the problem, and f∈Y represents external forcing or data. In the non-separable setting, the domain D(A) need not be dense in X, and continuity is generally understood with respect to weak or weak* topologies. Accordingly, solutions are defined in weak, weak*, or bi dual senses, depending on the analytic structure of A.

Elliptic-Type Equations in EBNS

Elliptic equations form the static backbone of PDE theory. In NSBS, they are treated as operator equations of the form εu = f, where ° :X→X\* (or X→Y) is typically coercive in an appropriate weak sense [1].

Parabolic-Type Equations and Evolution Operators

Parabolic PDEs correspond to time-dependent problems and are naturally modelled as abstract evolution equations:

General Evolution-Type Equations

Beyond parabolic equations, many systems of interest—particularly in nonlinear dynamics and inverse problems—take the form of general evolution equations:

Nonlinear Operators and Monotonicity in EBNS

Analytical Outlook Section

3 establishes a unified abstract framework for elliptic, parabolic, and evolution-type PDEs on non-separable Banach spaces. Section 4 will instantiate this framework in concrete non-separable spaces such as l∞ (Γ) and C(K). Section 5 will extend the analysis to inverse problems for nonlinear dynamics. Sections 6–7 will provide adapted existence and uniqueness theorems.

Constructive Examples in Canonical Non-Separable Banach Spaces

This section provides explicit examples of elliptic, parabolic, and evolution-type partial differential equations posed on canonical non-separable Banach spaces. The purpose is to illustrate how classical PDE phenomena persist—albeit in modified form—when separability is abandoned. Throughout, emphasis is placed on operator-theoretic formulations, weak and weak* solutions, and approximate compactness in place of classical compactness [6].

Elliptic Equations on l∞ (Γ)

Elliptic-Type PDEs on C(K) with Non-Metrisable K

Let K be a compact Hausdorff space that is not metrisable, and consider

This example demonstrates that elliptic-type behaviour is preserved even when the underlying topological space lacks any countable structure.

Parabolic Equations on l∞ (Γ)

Consider the abstract parabolic equation

Nonlinear Evolution Equations in Dual Spaces

Approximate Compact Perturbations in Concrete Models

Let A₀ be any of the operators above, and let

Summary and Transition

The examples developed in this section confirm that:

a. Elliptic and parabolic PDEs admit meaningful formulations on canonical NSBS. b. Weak and weak* solutions arise naturally and consistently.

c. Approximate compactness replaces classical compactness in a concrete, verifiable manner.

d. Operator-theoretic methods scale naturally to non-separable settings.

These constructions prepare the ground for section 5, where the same analytical machinery will be applied to inverse problems for nonlinear dynamics, including identifiability, stability, and regularisation in NSBS.

Theorem-Level Existence

Elliptic Existence on P∞via Diagonal Coercivity 

Elliptic Existence on C(K) with non-Metrisable K

Parabolic Existence on with Explicit Mild Solutions

Weak Existence for Dual-Valued Evolution in L

Perturbative Existence under Approximately Compact Terms

Abstract Parabolic Well-Posedness under Approximately Compact Resolvent Hypotheses

Inverse Problems for Nonlinear Dynamics in NSBS

This section develops a general framework for inverse problems associated with evolution equations posed on non-separable Banach spaces (NSBS). The analysis is built upon the parabolic well-posedness result of section 5.6 and exploits the approximate compactness of forward solution operators as the central analytical mechanism governing well-posedness, stability, and regularisation [3,10-12,16]. Throughout, we adopt the viewpoint that inverse problems are best understood as operator inversion problems in appropriate topological settings, rather than as pointwise reconstruction tasks.

Abstract Inverse Problem Setting

Parameter-to-State and Forward Operators

Well-Posedness and Approximate Compactness

In classical inverse problems on separable spaces, compactness of the forward operator is the fundamental source of well-posedness. In NSBS, compactness is typically unavailable; however, approximate compactness plays an analogous role [2,4,8-11,14,15]

Identifiability

Stability in Weak and Weak Topologies*

Regularisation in NSBS

Summary

Section 6 establishes a complete inverse-problem framework for nonlinear dynamics on NSBS:

i. The forward map inherits approximate compactness from parabolic smoothing.

ii. Well-posedness is intrinsic but structurally controlled.

iii. Identifiability, weak stability, and regularisation admit precise operator-theoretic formulations.

iv. Existence of regularised solutions follows without separability or compactness assumptions.

This prepares the ground for section 7, where adapted existence and uniqueness theorems and problem-specific refinements will be derived, and for concrete inverse-problem models tied to the examples of section 4.

Adapted Existence, Uniqueness, and Stability Results in NSBS

This section consolidates the abstract parabolic well-posedness theory with the inverse-problem framework developed earlier. We state final results that are intrinsic to non-separable Banach spaces, replacing classical compactness and strong stability by approximate compactness and weak/weak* stability, respectively. The theorems below are formulated so as to apply uniformly to the constructive models of section 4 and to the abstract parabolic setting with approximately compact resolvents [2,3,6,8,12,14,15].

Standing Assumptions

Forward Existence and Uniqueness (Uniform in the Parameter)

Approximate Compact Smoothing and Consequences

Adapted Uniqueness and Identifiability

Stability in Weak and Weak* Topologies

Existence and Stability of Regularised Solutions

Synthesis and ScopeThe results of this section establish that, in NSBS:

a. Forward problems are well-posed under uniform parabolic hypotheses.

b. Inverse problems are intrinsically well-posed but structurally controlled by approximate compactness.

c. Uniqueness and identifiability can be recovered under injectivity assumptions.

d. Stability is naturally formulated in weak or weak* topologies. e. Regularisation restores existence and convergence without separability or compactness.

Together, sections 5 to 7 provide a complete and internally consistent theory of PDE-driven inverse problems on non-separable Banach spaces, fully aligned with the operator-theoretic methodology of this work.

Applications and Perspectives

This section outlines representative applications of the theory developed in this article and delineates several directions for future research. The emphasis is on domains where non-separable Banach spaces arise intrinsically, rendering classical separable frameworks inadequate, and where approximate compactness provides the correct analytical substitute for compactness [2,3,5,6,8,10-12,15].

Applications

Infinite-Dimensional and Large-Scale Dynamical Systems.

Many dynamical systems with infinitely many degrees of freedom-such as continuum limits of large interacting agent models, mean-field limits with uncountable index sets, or distributed systems with heterogeneous parameters-naturally evolve in spaces like
or dual spaces of measures. In such contexts, separability is neither natural nor desirable.

The parabolic well-posedness theory with approximately compact resolvents (theorem 6.6.2) provides a rigorous foundation for: forward evolution, parameter-to-state mappings and inverse identification of heterogeneous coefficients, without requiring truncation to separable submodels.

Inverse Problems with Weak or Aggregate Observations.

In many inverse problems-particularly in control, economics, and networked systems-observations are weak, aggregated, or dual-valued. Typical examples include: averaged measurements, integral observations, weak* data arising from dual pairings. The framework developed here accommodates such situations naturally by: formulating inverse problems in weak and weak* topologies, establishing identifiability and stability without norm convergence, and guaranteeing existence of regularised solutions under minimal assumptions.

This is especially relevant when the state space is  or other dual NSBS.

PDE-Constrained Optimisation in Non-Separable Settings.

PDE-constrained optimisation problems often involve admissible sets of controls or coefficients that are bounded but non-separable. The regularisation theory developed in sections 5 and 7 yields: existence of minimisers for Tikhonov-type functionals, weak or weak* stability under noisy data, and convergence of regularised solutions without compactness. These results provide a rigorous analytical basis for optimisation problems that cannot be treated within classical Hilbert or separable Banach frameworks.

Functional Models Beyond Classical Sobolev Theory.

In applications where Sobolev embeddings fail or are unavailable-such as models defined on non-metrisable spaces or indexed by uncountable sets-the present approach offers an alternative route: (i) PDEs are formulated as operator equations; (ii) coercivity, accretivity, and monotonicity replace embedding-based arguments; (iii) and approximate compactness replaces compactness of embeddings.

This opens the door to PDE modelling in settings previously regarded as analytically inaccessible.

Methodological Perspectives

One of the central conceptual messages of this article is that approximate compactness is not merely a technical workaround, but a structural principle that: captures the essential analytical effects of compactness, remains meaningful in non-separable spaces, and governs well-posedness, stability, and regularisation in inverse problems.

Future work may further refine this notion, exploring its relationships with measures of non-compactness, entropy numbers, and quantitative stability estimates. '

While the present work focuses primarily on linear and semilinear parabolic equations, the framework extends naturally to: fully nonlinear evolution equations, differential inclusions, and variational inequalities on NSBS.

In such settings, weak monotonicity and accretivit-rather than linearity-are expected to play the decisive role in existence and uniqueness.

The approximate compactness of resolvents and semigroups suggests the possibility of a generalised spectral theory for parabolic operators on NSBS. Potential research directions include: asymptotic behaviour of solutions, spectral decompositions modulo approximate compactness, and long-time dynamics in weak or weak* topologies. Such developments would parallel classical spectral theory while remaining faithful to non-separable structures.

Although this article is purely analytical, the results have clear implications for computation: weak and weak* convergence are often more accessible numerically than strong convergence, regularisation in NSBS may be implemented via dual or variational formulations, and approximate compactness suggests natural discretisation strategies based on bounded approximations rather than finite-dimensional truncations.

Bridging the gap between this theory and numerical schemes is an important avenue for future research.

Outlook

The theory developed in this article demonstrates that partial differential equations and inverse problems can be treated rigorously in NSBS, without recourse to separable reductions or artificial compactness assumptions. By combining operator-theoretic formulations, approximate compactness, and weak/weak* analysis, we obtain a coherent framework that is both mathematically robust and adaptable to real-world modelling requirements. Future work will aim to: (a) extend the theory to broader classes of operators and dynamics, (b) refine stability and convergence results, (c) and explore interdisciplinary applications where non-separability is intrinsic rather than incidental.

Conclusion

This article has developed a unified functional–analytic framework for partial differential equations and inverse problems posed on non-separable Banach spaces. By abandoning separability as a foundational assumption and replacing classical compactness with approximate compactness, we have established existence, uniqueness, and stability results that remain valid in genuinely non-separable settings.

Elliptic, parabolic, and evolution-type equations were formulated intrinsically as operator equations on non-separable Banach spaces, with weak and weak* topologies treated as structural components rather than auxiliary tools. The resulting well-posedness theory provides a robust forward framework that supports the analysis of inverse problems without relying on compact embeddings or spectral decompositions.

For inverse problems, approximate compactness emerges as the key mechanism governing well-posedness, weak stability, and regularisation. Identifiability, weak and weak* convergence of reconstructions, and existence of regularised solutions were obtained under minimal and natural assumptions. A fully worked canonical example on l∞ (Γ) demonstrated that the abstract theory is both non-vacuous and operational in concrete non-separable models.

Overall, the results presented here show that non-separable Banach spaces can be treated as first-class analytical environments for PDEs and inverse problems. The framework opens new perspectives for modelling and analysis in contexts where separability fails intrinsically, and provides a foundation for further developments in nonlinear dynamics, spectral theory, and applied inverse problems beyond the separable paradigm.

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