Banach–Mazur theorem

In functional analysis, a field of mathematics, a key object of study is a normed space, which is a vector space equipped with a norm, which allows vectors to be measured. When they are infinite dimensional, normed spaces can be very complicated. A standard normed space is the space of continuous functions on the unit interval,, which is equipped with the norm, the maximum value of the function.
The Banach–Mazur theorem is a theorem that provides one way of bounding the complexity of certain well-behaved normed spaces. It states that every such normed space can be embedded into the normed space, in such a way that the norm of every vector is preserved. It is named after Stefan Banach and Stanisław Mazur.

Statement

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 : 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 intersects only at : thus the space of smooth functions is isometrically isomorphic to a space of nowhere-differentiable functions. Note that the space of smooth functions is dense in.