Английская Википедия:Banach–Mazur theorem
Шаблон:Distinguish In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur.
Statement
Every real, separable Banach space Шаблон:Math is isometrically isomorphic to a closed subspace of Шаблон:Math, 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 Шаблон:Math 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 Шаблон:Math, but for every Banach space Шаблон:Mvar, one can find a compact Hausdorff space Шаблон:Mvar and an isometric linear embedding Шаблон:Mvar of Шаблон:Mvar into the space Шаблон:Math of scalar continuous functions on Шаблон:Mvar. The simplest choice is to let Шаблон:Mvar be the unit ball of the continuous dual Шаблон:Math, equipped with the w*-topology. This unit ball Шаблон:Mvar is then compact by the Banach–Alaoglu theorem. The embedding Шаблон:Mvar is introduced by saying that for every Шаблон:Math, the continuous function Шаблон:Math on Шаблон:Mvar is defined by
- <math> \forall x' \in K: \qquad j(x)(x') = x'(x).</math>
The mapping Шаблон:Mvar 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 Шаблон:Mvar is isometric to a subspace of Шаблон:Math, the space of real continuous functions on the product of Шаблон:Mvar copies of the unit interval.
Stronger versions of the theorem
Let us write Шаблон:Math for Шаблон:Math. In 1995, Luis Rodríguez-Piazza proved that the isometry Шаблон:Math can be chosen so that every non-zero function in the image Шаблон:Math is nowhere differentiable. Put another way, if Шаблон:Math consists of functions that are differentiable at at least one point of Шаблон:Math, then Шаблон:Mvar can be chosen so that Шаблон:Math This conclusion applies to the space Шаблон:Math itself, hence there exists a linear map Шаблон:Math that is an isometry onto its image, such that image under Шаблон:Mvar of Шаблон:Math (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects Шаблон:Mvar only at Шаблон:Math: 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 Шаблон:Math.
References
Шаблон:Banach spaces Шаблон:Functional Analysis
