Banach-Mazur theorem

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In mathematics, the Banach-Mazur theorem is a theorem of functional analysis. Very roughly, it states that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur.

[edit] Statement of the theorem

Every separable Banach space is isometrically isomorphic to a closed subspace of C⁰([0, 1]; R), the space of all continuous functions from the unit interval into the real line.

[edit] Comments

On the one hand, the Banach-Mazur theorem seems to tell us that the seemingly vast collection of all separable Banach spaces is not that vast or difficult to work with, since a separable Banach space is "just" a collection of continuous paths. On the other hand, the theorem tells us that C⁰([0, 1]; R) is a "really big" space, big enough to contain every possible separable Banach space.

[edit] Reference

• Bessaga, Czesław, & Pełczyński, Alexsander (1975). Selected topics in infinite-dimensional topology. Warszawa: PWN.