Banach & Topological Vector Spaces

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Abstractions about vector spaces equipped with norms, metrics, or order structures — Banach, Fréchet, Riesz, and sequence spaces — and the convergence, compactness, and integration notions, such as the Bochner integral, weak compactness, and absolute or normal convergence, that characterize functions and functionals on them.

30 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.

• Absolute convergence — Convergence of a series or improper integral after replacing every summand or integrand value by its absolute magnitude.
• Aubin–Lions lemma — A compactness result for time-dependent functions combining spatial compact embedding with control of a time derivative in a weaker space.
• Ba space — The Banach space of bounded finitely additive signed measures on an algebra of sets, equipped with the total-variation norm.
• Banach–Mazur compactum — The compact metric space of isometry classes of fixed-dimensional normed spaces under logarithmic Banach–Mazur distance.
• BK-space — A Banach sequence space in which every coordinate projection is continuous.
• Bochner integral — The Banach-space-valued extension of the Lebesgue integral defined by norm limits of integrals of simple functions.
• Differentiable vector-valued functions from Euclidean space — Differentiable maps from an open Euclidean domain into a topological vector space, with derivatives encoded by continuous multilinear maps.
• Eberlein–Šmulian theorem — The Banach-space result equating weak compactness with weak sequential compactness and weak limit-point compactness.
• F-space — A real or complex vector space equipped with a complete translation-invariant metric whose addition and scalar multiplication are continuous.
• Gelfand–Shilov space — A space of smooth test functions whose derivatives and polynomially weighted values obey factorial growth bounds controlling simultaneous decay and regularity.
• Kōmura's theorem — An almost-everywhere differentiability theorem for absolutely continuous functions taking values in a reflexive Banach space.
• L-infinity — The Banach space of essentially bounded measurable functions under the essential-supremum norm, with bounded sequences as the counting-measure case.
• L-semi-inner product — A Banach-space generalization of inner product that is linear in one argument and positive but need not be conjugate symmetric or additive in the other.
• Lebesgue's lemma — An approximation bound stating that a bounded linear projection's error is at most one plus its operator norm times the best attainable error from the target subspace.
• Locally integrable function — A measurable function whose absolute value has finite integral on every compact subset of its open domain, without requiring finite integral over the whole domain.
• Marcinkiewicz interpolation theorem — An interpolation theorem deriving strong intermediate Lp bounds for a sublinear operator from suitable weak-type endpoint bounds.
• Mazur's lemma — A result stating that convex combinations of tails of a weakly convergent sequence in a normed space can be chosen to converge in norm to the same limit.
• McShane integral — A gauge integral using free tagged partitions whose tags may lie outside their associated subintervals, yielding for real-valued functions exactly the Lebesgue-integrable class.
• Microcontinuity — The nonstandard-analysis form of continuity requiring infinitely close inputs to have infinitely close outputs.
• Normal convergence — Convergence of a function series whose sum of termwise uniform norms is finite.
• Nowhere continuous function — A function that fails the continuity condition at every point of its domain.
• Order convergence — Convergence in an ordered vector lattice defined by eventual confinement between bounds that close monotonically on the limit.
• Projective tensor product — The tensor product of locally convex spaces equipped with the strongest locally convex topology making the canonical bilinear map continuous.
• Riesz space — A real vector space equipped with a lattice order compatible with vector addition and nonnegative scalar multiplication.
• Riesz–Markov–Kakutani representation theorem — A representation theorem identifying continuous linear functionals on suitable spaces of continuous functions with integration against unique regular measures.
• 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.
• Smith space — A complete compactly generated locally convex space possessing one compact set that absorbs every compact subset.
• Strongly positive bilinear form — A bilinear form on a normed vector space that dominates a fixed positive multiple of squared norm on every vector.
• Uniform norm — The supremum of pointwise magnitudes of a bounded function, inducing the metric of uniform convergence and the maximum-coordinate norm in finite dimensions.