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.
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[edit] Statement of the theorem
Every real, separable Banach space (X, || ||) 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] Stronger versions of the theorem
In 1995, Luis Rodríguez-Piazza proved that the isometry i : X → C0([0, 1]; R) can be chosen so that every non-zero function in the image i(X) is nowhere differentiable. Put another way, if D denotes the subset of C0([0, 1]; R) consisting of those functions that are differentiable at at least one point of [0, 1], then i can be chosen so that i(X) ∩ D = {0}. This conclusion applies to the space C0([0, 1]; R) itself, leading to the seemingly paradoxical result that there exists a linear map i from C0([0, 1]; R) to itself that is an isometry onto its image, such that image under i of C1([0, 1]; R) (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects D only at 0: a space of smooth functions can be isometrically isomorphic to a space of nowhere-differentiable functions!
[edit] References
- Bessaga, Czesław, & Pełczyński, Alexsander (1975). Selected topics in infinite-dimensional topology. Warszawa: PWN.
- Rodríguez-Piazza, Luis (1995). "Every separable Banach space is isometric to a space of continuous nowhere differentiable functions". Proc. Amer. Math. Soc. 123: 3649–3654. doi:.
