Banach–Mazur theorem

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.

tatement of the theorem

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

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 "C"⁰( [0, 1] ; R) is a "really big" space, big enough to contain every possible separable Banach space.

tronger versions of the theorem

In 1995, Luis Rodríguez-Piazza proved that the isometry "i" : "X" → "C"⁰( [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 "C"⁰( [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 "C"⁰( [0, 1] ; R) itself, leading to the seemingly paradoxical result that there exists a linear map "i" from "C"⁰( [0, 1] ; R) to itself that is an isometry onto its image, such that image under "i" of "C"¹( [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!

References

* cite book
author=Bessaga, Czesław, & Pełczyński, Alexsander
title=Selected topics in infinite-dimensional topology
location=Warszawa
publisher=PWN
year=1975
* cite journal
last = Rodríguez-Piazza
first = Luis
title = Every separable Banach space is isometric to a space of continuous nowhere differentiable functions
journal = Proc. Amer. Math. Soc.
volume = 123
year = 1995
number = 12
pages = 3649–3654
doi = 10.2307/2161889