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Localized locally convex topologies

Thierry De Pauw

Abstract

Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a family $\mathcal{C}$ of convex subsets of $X$. The distributions $F$ arising as $\mathrm{div} (v)$ are expected to be the members of the dual of well-chosen $X$ with respect to an appropriate localized topology $\mathcal{T}_{\mathcal{C}}$. In this work, the emphasis is on studying the functional analytic properties of $\mathcal{T}_{\mathcal{C}}$, according to those of $\mathcal{T}$ and $\mathcal{C}$. For instance, we show that in all foreseen applications, $\mathcal{T}_{\mathcal{C}}$ is sequential but none of Fréchet-Urysohn, barrelled, and bornological. These awkward phenomena are illustrated explicitly on a specific example corresponding to the distributional divergence of continuous vector fields in $\mathbb{R}^m$. We also show that, essentially, $\mathcal{T}_{\mathcal{C}}$ is semireflexive if and only if members of $\mathcal{C}$ are $\mathcal{T}$-compact. This leads to an abstract existence theorem, thereby establishing a general scheme for characterizing those $F$ such that $\mathrm{div} (v) = F$ for various classes of regularity of $v$, various classes of domains, and various boundary conditions.

Localized locally convex topologies

Abstract

Motivated by ill-posed PDEs such as we study locally convex topologies on real vector spaces that are a ``localized'' version of a locally convex topology to members of a family of convex subsets of . The distributions arising as are expected to be the members of the dual of well-chosen with respect to an appropriate localized topology . In this work, the emphasis is on studying the functional analytic properties of , according to those of and . For instance, we show that in all foreseen applications, is sequential but none of Fréchet-Urysohn, barrelled, and bornological. These awkward phenomena are illustrated explicitly on a specific example corresponding to the distributional divergence of continuous vector fields in . We also show that, essentially, is semireflexive if and only if members of are -compact. This leads to an abstract existence theorem, thereby establishing a general scheme for characterizing those such that for various classes of regularity of , various classes of domains, and various boundary conditions.
Paper Structure (16 sections, 29 theorems, 106 equations)

This paper contains 16 sections, 29 theorems, 106 equations.

Table of Contents

  1. Existence and uniqueness of a localization
  2. General observations
  3. Countable localizing families of closed convex sets
  4. Countable localizing families of compact convex sets
  5. First dual, dense subspaces, and extensions
  6. Bounded weak*, Banach-Grothendieck, and Krein-Šmulian
  7. Second dual, semireflexivity, and local base
  8. Abstract existence theorem
  9. Abstract compactness theorem
  10. One example: The divergence of continuous vector fields
  11. Sequential vs. Fréchet-Urysohn topological spaces
  12. Locally convex topological vector spaces
  13. Dual systems and the Mackey-Arens theorem
  14. Barrelled spaces
  15. Bornological spaces
  16. ...and 1 more sections

Key Result

Theorem 1.4

Assume $X[\mathscr{T}]$ is a locally convex topological vector space and $\mathscr{C}$ is a localizing family in $X$. There then exists a unique localization of $\mathscr{T}$ by $\mathscr{C}$ and it is given by the formula in EUL.6 below.

Theorems & Definitions (120)

  • Theorem 1.4
  • Remark 1.5
  • proof : Proof of uniqueness
  • proof : Proof of condition \ref{['EUL.3']}(i)
  • proof : Proof of condition \ref{['EUL.3']}(ii)
  • proof
  • proof
  • proof
  • proof : Proof that addition of vectors in $X$ is $\mathscr{S}$-continuous
  • proof
  • ...and 110 more