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.

Statement of the theorem
Every real, separable Banach space  is isometrically isomorphic to a closed subspace of , 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 "only" a collection of continuous paths. On the other hand, the theorem tells us that  is a "really big" space, big enough to contain every possible separable Banach space.

Non-separable Banach spaces cannot embed isometrically in the separable space , but for every Banach space , one can find a compact Hausdorff space  and an isometric linear embedding  of  into the space  of scalar continuous functions on . The simplest choice is to let  be the unit ball of the continuous dual , equipped with the w*-topology. This unit ball  is then compact by the Banach–Alaoglu theorem. The embedding  is introduced by saying that for every , the continuous function  on  is defined by

The mapping  is linear, and it is isometric by the Hahn–Banach theorem.

Another generalization was given by Kleiber and Pervin (1969): a metric space of density equal to an infinite cardinal  is isometric to a subspace of , the space of real continuous functions on the product of  copies of the unit interval.

Stronger versions of the theorem
Let us write  for . In 1995, Luis Rodríguez-Piazza proved that the isometry  can be chosen so that every non-zero function in the image  is nowhere differentiable. Put another way, if  consists of functions that are differentiable at at least one point of , then  can be chosen so that  This conclusion applies to the space  itself, hence there exists a linear map  that is an isometry onto its image, such that image under  of  (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects  only at : thus the space of smooth functions (with respect to the uniform distance) is isometrically isomorphic to a space of nowhere-differentiable functions. Note that the (metrically incomplete) space of smooth functions is dense in .

References
 
 
 

Category:Continuous mappings
Category:Theorems in functional analysis