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A study of some classes of topological vector spaces

Glen Skanes

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Abstract

In this dissertation, we study two important classes of locally convex spaces in great detail. The first one is the class of (DF)-spaces, introduced by GROTHENDIECK as a prototype of duals of (F)-spaces. A non-locally convex analogue of these spaces is also discussed. The second one is the class of Schwartz spaces. The role of certain Banach spaces as "universal" to the "variety" of all Schwartz spaces in investigated. Finally, co-Schwartz spaces are introduced in a fashion analogous to the study of co-nuclear spaces from that of nuclear spaces. Several counter-examples are provided.

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In this dissertation, we study two important classes of locally convex spaces in great detail. The first one is the class of (DF)-spaces, introduced by GROTHENDIECK as a prototype of duals of (F)-spaces. A non-locally convex analogue of these spaces is also discussed. The second one is the class of Schwartz spaces. The role of certain Banach spaces as "universal" to the "variety" of all Schwartz spaces in investigated. Finally, co-Schwartz spaces are introduced in a fashion analogous to the study of co-nuclear spaces from that of nuclear spaces. Several counter-examples are provided.

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Available abstract

In this dissertation, we study two important classes of locally convex spaces in great detail. The first one is the class of (DF)-spaces, introduced by GROTHENDIECK as a prototype of duals of (F)-spaces. A non-locally convex analogue of these spaces is also discussed. The second one is the class of Schwartz spaces. The role of certain Banach spaces as "universal" to the "variety" of all Schwartz spaces in investigated. Finally, co-Schwartz spaces are introduced in a fashion analogous to the study of co-nuclear spaces from that of nuclear spaces. Several counter-examples are provided.

Key concepts: Locally convex topological vector space, Topological tensor product, Dual polyhedron, Mathematics, Fréchet space, Interpolation space, Birnbaum–Orlicz space, Pure mathematics

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