Topological Vector Spaces & Bundles

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Abstractions about locally convex spaces, manifolds, bundles, tensor products, diffeomorphisms, homomorphisms, and reflexivity or completeness conditions.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Banach bundle — A fiber bundle whose fibers are Banach spaces and whose local trivializations and transition maps preserve compatible continuous linear structure.
• Fréchet manifold — Build an infinite-dimensional smooth manifold from charts valued in Fréchet spaces, while making the chosen smooth calculus and the failure of general Banach inverse-function machinery explicit.
• Local diffeomorphism — Map smooth manifolds so that every source point has a neighborhood carried diffeomorphically onto an open target neighborhood, without requiring global injectivity.
• Projective tensor product — The tensor product of locally convex spaces equipped with the strongest locally convex topology making the canonical bilinear map continuous.
• Schwartz topological vector space — A locally convex topological vector space whose bounded sets are precompact, equivalently whose neighborhoods satisfy a finite-covering condition after suitable shrinking and scaling.
• Semi-reflexive space — A locally convex topological vector space whose canonical map into its strong bidual is algebraically onto, without necessarily being a topological isomorphism.
• Topological homomorphism — A continuous linear map between topological vector spaces that induces a topological isomorphism from the quotient by its kernel onto its image.
• Webbed space — A topological vector space equipped with a web structure that supports generalized closed-graph and open-mapping theorems.