Banach–Mazur theorem

Every separable Banach space is linearly isometric to a closed subspace of the Banach space C([0,1]) of continuous real-valued functions on the unit interval, equipped with the supremum norm.

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Separable Banach space: A Banach space containing a countable dense subset. Separability supplies the countable dense structure needed to build the interval-valued representation.
Banach space: A complete normed vector space. The theorem applies to complete normed spaces, not arbitrary normed spaces.
Stefan Banach: A Polish mathematician who helped establish functional analysis as a distinct field. Banach’s work on abstract normed spaces forms the theorem’s mathematical setting.
Banach–Mazur compactum: A compact space representing finite-dimensional normed spaces up to linear isometry. It develops the geometric classification viewpoint suggested by universal representation results.
Continuous dual space: The space of continuous linear functionals on a normed vector space. The embedding encodes vectors through evaluations by suitably chosen functionals.
Normed vector space: A vector space equipped with a norm that measures vector size. Banach spaces and their isometries are defined using this underlying structure.
Stanisław Mazur: A Polish mathematician known for contributions to functional analysis and convexity. Mazur co-authored the foundational text associated with this representation result.
Universal space: A space into which every member of a specified class embeds in a prescribed way. C([0,1]) is universal for separable Banach spaces under linear isometric embedding.
Hahn–Banach theorem: A theorem extending bounded linear functionals from subspaces without increasing their norm. It provides functionals that detect vector norms, enabling an isometric representation.
Dense subset: A subset whose closure is the entire space. A countable dense subset is the defining approximation condition in separability.