In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Banach–Mazur theorem is a theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 of functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

. Very roughly, it states that most well-behaved

Well-behaved

Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...

 normed spaces are subspace

Linear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

s of the space of continuous paths

Path (topology)

In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...

. It is named after Stefan Banach

Stefan Banach

Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....

 and Stanisław Mazur.

Statement of the theorem

Every real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

, separable Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

 (X, || ||) is isometrically isomorphic

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 to a closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 subspace of C ⁰([0, 1]; R), the space of all continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

s from the unit interval

Interval

Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement...

 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 "just" a collection of continuous paths. On the other hand, the theorem tells us that C ⁰([0, 1]; R) is a "really big" space, big enough to contain every possible separable Banach space.

Stronger versions of the theorem

Let's write  C ^k[0, 1]  for  C ^k([0, 1]; R).  In 1995, Luis Rodríguez-Piazza proved that the isometry i : X → C ⁰[0, 1] can be chosen so that every non-zero function in the image

Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

 i(X) is nowhere differentiable. Put another way, if D denotes the subset of C ⁰[0, 1] consisting of those functions that are differentiable at least one point of [0, 1], then i can be chosen so that i(X) ∩ D = {0}. This conclusion applies to the space C ⁰[0, 1] itself, hence there exists a linear map  i  from C ⁰[0, 1] to itself that is an isometry onto its image, such that image under i of C ¹[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 (w.r. 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 C ⁰[0, 1].