Banach–Mazur theorem

Template:Distinguish 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 $C([0,1],ℝ)$ of continuous functions on the unit interval, $f:[0,1]\to ℝ$, which is equipped with the norm $\Vert f\Vert =\underset{x\in [0,1]}{max}|f(x)|$, 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 (separable). It states that every such normed space can be embedded into the normed space $C([0,1],ℝ)$, in such a way that the norm (length) of every vector is preserved (a property known as an isometry). It is named after Stefan Banach and Stanisław Mazur.

Statement

Every real, separable Banach space Template:Math is isometrically isomorphic to a closed subspace of Template: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 Template: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 Template:Math, 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 Template:Math of scalar continuous functions on Template:Mvar. The simplest choice is to let Template:Mvar be the unit ball of the continuous dual Template:Math, 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 Template:Math, the continuous function Template:Math 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 Template:Math, 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 Template:Math for Template:Math. In 1995, Luis Rodríguez-Piazza proved that the isometry Template:Math can be chosen so that every non-zero function in the image Template:Math is nowhere differentiable. Put another way, if Template:Math consists of functions that are differentiable at at least one point of Template:Math, then Template:Mvar can be chosen so that Template:Math This conclusion applies to the space Template:Math itself, hence there exists a linear map Template:Math that is an isometry onto its image, such that image under Template:Mvar of Template:Math (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects Template:Mvar only at Template: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 Template:Math.

References

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