Banach–Mazur theorem

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

${\displaystyle \forall x'\in K:\qquad j(x)(x')=x'(x).}$

The mapping j 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 α is isometric to a subspace of C⁰([0,1]^α, R), the space of real continuous functions on the product of α 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 i 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 i of C⁰[0, 1] (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects D 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].

References[]

• Bessaga, Czesław & Pełczyński, Aleksander (1975). Selected topics in infinite-dimensional topology. Warszawa: PWN.
• Kleiber, Martin; Pervin, William J. (1969). "A generalized Banach-Mazur theorem". Bull. Austral. Math. Soc. 1 (2): 169–173. doi:10.1017/S0004972700041411 – via Cambridge University Press.
• Rodríguez-Piazza, Luis (1995). "Every separable Banach space is isometric to a space of continuous nowhere differentiable functions". Proc. Amer. Math. Soc. American Mathematical Society. 123 (12): 3649–3654. doi:10.2307/2161889. JSTOR 2161889.