Banach–Mazur theorem

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 (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.

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 C⁰([0, 1], R) 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 C⁰([0, 1], R), but for every Banach space Template:Mvar, one can find a compact Hausdorff space Template:Mvar and an isometric linear embedding Template:Mvar of Template:Mvar into the space C(K) of scalar continuous functions on Template:Mvar. The simplest choice is to let Template:Mvar be the unit ball of the continuous dual X ′, equipped with the w*-topology. This unit ball Template:Mvar is then compact by the Banach–Alaoglu theorem. The embedding Template:Mvar is introduced by saying that for every x ∈ X, the continuous function j(x) on Template:Mvar is defined by

$\mathrm{\forall }{x}^{\prime }\in K:j(x)(x^{\prime })=x^{\prime }(x).$

The mapping Template: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 Template:Mvar is isometric to a subspace of C⁰([0,1]^α, R), the space of real continuous functions on the product of Template:Mvar copies of the unit interval.

Stronger versions of the theorem

Let us write C^k[0, 1] for C^k([0, 1], R). In 1995, Luis Rodríguez-Piazza proved that the isometry i : X → C⁰[0, 1] can be chosen so that every non-zero function in the image i(X) is nowhere differentiable. Put another way, if D ⊂ C⁰[0, 1] consists of functions that are differentiable at at least one point of [0, 1], then Template:Mvar can be chosen so that i(X) ∩ D = {0}. This conclusion applies to the space C⁰[0, 1] itself, hence there exists a linear map i : C⁰[0, 1] → C⁰[0, 1] that is an isometry onto its image, such that image under Template:Mvar of C⁰[0, 1] (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects Template:Mvar only at 0: 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 C⁰[0, 1].