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Sven-Ake Wegner

Publications and source records attributed to Sven-Ake Wegner.

At least 19 recordsLinked to original sources
arXiv2026

Condensed mathematics through compactological spaces

In their 2022 lecture notes on condensed sets, Clausen and Scholze mentioned in a remark that the important subclass of quasiseparated condensed sets is equivalent to the category of so-called compactological spaces defined by Waelbroeck in the 1960s. In this paper we study the latter category in detail, give a rigorous proof of Clausen and Scholze's claim, and establish that condensed sets are a formal categorical completion of Waelbroeck's compactological spaces. The latter answers a question asked by Hanson in 2023 and permits the interpretation of compactological sets as an `elementary' approach to condensed mathematics.

math.FA↗
arXiv2026

A homological approach to (Grothendieck's) completeness problem for regular LB-spaces

We consider the long-standing question of whether every regular LB-space is complete. This problem has been open since the 1950s and originates in Grothendieck's early work in functional analysis. Rather than seeking a direct proof or counterexample, our approach is to study weak versions of the problem using homological methods. We consider the categories of complete and, respectively, regular LB-spaces, establish that their derived categories are well-defined with respect to several exact structures, and show that there are canonical triangle functors between them. If one of these functors were not an equivalence, this would provide a negative answer to Grothendieck's question. In contrast, we prove that one of them is an equivalence. This may be interpreted as evidence in favor of an affirmative answer to the original problem, and it shows in particular that the two classes of spaces share the same homological algebra, even if they were to differ.

math.FA↗
arXiv2025

Bornological LB-spaces and idempotent adjunctions

The notion of an LB-space was introduced by Grothendieck in his 1953 thèse, referring to a countable colimit of Banach spaces taken within the category of locally convex topological vector spaces, and refining prior work done by Dieudonné, Schwartz and Köthe. Recently, two different notions of `bornological LB-spaces' emerged: one, given by Stempfhuber, refers to countable colimits of Banach spaces as well, but now taken in the category of bornological vector spaces. The other one, given by Bambozzi, Ben-Bassat and Kremnizer, refers to bornologifications of regular LB-spaces, i.e., of LB-spaces in the Grothendieck sense having the additional property that every bounded subset of the colimit is contained and bounded in one of its Banach steps. In this note, we show that the two notions are distinct, but nevertheless closely related. This involves, in particular, an intimate study of the idempotent adjunction of the bornologification and topologification functors.

math.FA↗
arXiv2024

Lecture notes on high-dimensional data

These are lecture notes based on the first part of a course on 'Mathematical Data Science', which I taught to final year BSc students in the UK in 2019-2020. Topics include: concentration of measure in high dimensions; Gaussian random vectors in high dimensions; random projections; separation/disentangling of Gaussian data. A revised version has been published as part of the textbook [Mathematical Introduction to Data Science, Springer, Berlin, Heidelberg, 2024, https://link.springer.com/book/10.1007/978-3-662-69426-8].

math.FA↗
arXiv2024

The left heart and exact hull of an additive regular category

Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $\mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $\mathcal{E}$, and can be constructed as the heart $\mathcal{LH}(\mathcal{E})$ of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$ or as the localization of the category of monomorphisms in $\mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $\operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. Additive regular categories can be characterized as those subcategories of abelian categories which are closed under subobjects. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $\mathcal{E}$ can be found as the heart of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$, or as the localization of the category of monomorphisms of $\mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.

math.CT↗
arXiv2022

The category of Silva spaces is not integral

We establish that the category of Silva spaces, aka LS-spaces, formed by countable inductive limits of Banach spaces with compact linking maps as objects and linear and continuous maps as morphisms, is not an integral category. The result carries over to the category of PLS-spaces, i.e., countable projective limits of LS-spaces -- which contains prominent spaces of analysis such as the space of distributions and the space of real analytic functions. As a consequence, we obtain that both categories neither have enough projective nor enough injective objects. All results hold true when 'compact' is replaced by 'weakly compact' or 'nuclear'. This leads to the categories of PLS-, PLS$_{\text{w}}$- and PLN-spaces, which are examples of 'inflation exact categories with admissible cokernels' as recently introduced by Henrard, Kvamme, van Roosmalen and the second-named author.

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arXiv2022

A Lumer-Phillips type generation theorem for bi-continuous semigroups

The famous 1960s Lumer-Phillips Theorem states that a closed and densely defined operator $A\colon D(A)\subseteq X\rightarrow X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,D(A))$ is dissipative and the range of $λ-A$ is surjective in $X$ for some $λ>0$. In this paper, we establish a version of this result for bi-continuous semigroups and apply the latter amongst other examples to the transport equation as well as to flows on infinite networks.

math.FA↗
arXiv2020

Examples and non-examples of integral categories and the admissible intersection property

Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories, and that there exist semi-abelian categories that are neither integral nor quasi-abelian. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense considered recently by Br{ü}stle, Hassoun and Tattar. This exhibits that a rich class of non-abelian categories having this property arises naturally in functional analysis.

math.CT↗
arXiv2020

On a counterexample in connection with the Picard-Lindelöf theorem

We give an example, which demonstrates that in the situation of the Picard-Lindelöf theorem, the Lipschitz condition on the right hand side $f(x,y)$ with respect to $y$, cannot be replaced by Lipschitz continuity in $y$ for every $x$. We show that, in our example, the classical Euler method detects only one of infinitely many solutions and we outline how the latter can be adjusted to find also other solutions numerically.

math.CA↗
arXiv2020

Dissipative extensions and port-Hamiltonian operators on networks

In this article we study port-Hamiltonian partial differential equations on certain one-dimensional manifolds. We classify those boundary conditions that give rise to contraction semigroups. As an application we study port-Hamiltonian operators on networks whose edges can have finite or infinite length. In particular, we discuss possibly infinite networks in which the edge lengths can accumulate zero and port-Hamiltonian operators with Hamiltonians that neither are bounded nor bounded away from zero. We achieve this, by first providing a new description for maximal dissipative extensions of skew-symmetric operators. The main technical tool used for this is the notion of boundary systems. The latter generalizes the classical notion of boundary triple(t)s and allows to treat skew-symmetric operators with unequal deficiency indices. In order to deal with fairly general variable coefficients, we develop a theory of possibly unbounded, non-negative, injective weights on an abstract Hilbert space.

math.FA↗
arXiv2019

Engaging students in conjecturing through homework in Real Analysis and Differential Equations

In this note we report on an implementation of discovery-oriented problems in courses on Real Analysis and Differential Equations. We explain a type of task-design that gives students the opportunity to conjecture, refute and prove. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework assignments. In addition to several concrete examples we also discuss feedback and assessment outcomes of our students.

math.HO↗
arXiv2019

Well-posedness of a class of hyperbolic partial differential equations on the semi-axis

In this article we study a class of hyperbolic partial differential equations of order one on the semi-axis. The so-called port-Hamiltonian systems cover for instance the wave equation and the transport equation, but also networks of the aforementioned equations fit into this framework. Our main results firstly characterize the boundary conditions which turn the corresponding linear operator into the generator of a strongly continuous semigroup. Secondly, we equip the equation with inputs (control) and outputs (observation) at the boundary and prove that this leads to a well-posed boundary control system. We illustrate our results via an example of coupled transport equations on a network, that allows to model transport from and to infinity. Moreover, we study a vibrating string of infinite length with one endpoint. Here, we show that our results allow to treat cases where the physical constants of the string tend to zero at infinity.

math.FA↗
arXiv2018

An outlook on self-assessment of homework assignments in higher mathematics education

We discuss first experiences with a new variant of self-assessment in higher mathematics education. In our setting, the students of the course have to mark a part of their homework assignments themselves and they receive the corresponding credit without that any later changes are carried out by the teacher. In this way we seek to correct the imbalance between student-centered learning arrangements and assessment concepts that keep the privilege to grade (or mark) completely with the teacher. We present results in the form of student feedback from a course on functional analysis for 3rd and 4th year students. Moreover we analyze marking results from two courses on real analysis. Here, we compare tasks marked by the teacher and tasks marked by the students.

math.HO↗
arXiv2018

Some remarks on the notions of boundary systems and boundary triple(t)s

In this note we show that if a boundary system in the sense of (Schubert et al. 2015) gives rise to any skew-self-adjoint extension, then it induces a boundary triplet and the classification of all extensions given by (Schubert et al. 2015) coincides with the skew-symmetric version of the classical characterization due to (Gorbachuk et al. 1991). On the other hand we show that for every skew-symmetric operator there is a natural boundary system which leads to an explicit description of at least one maximal dissipative extension. This is in particular also valid in the case that no boundary triplet exists for this operator.

math.FA↗
arXiv2017

Pivot duality of universal interpolation and extrapolation spaces

It is a widely used method, for instance in perturbation theory, to associate with a given C0-semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L^2(R), the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff, already complete, and can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space.

math.FA↗
arXiv2017

The heart of the Banach spaces

Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beilinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel-cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.

math.CT↗
arXiv2016

An open day in the metric space

We report on a workshop for grade eleven high school students, which took place in the framework of a university open day. During the workshop the participants first discovered the key properties of the intuitive concept of distance from real life examples. After this preparation, the formal definition of a metric space was introduced and discussed in small groups by means of problem-oriented exercise sessions.

math.HO↗