Banach–Mazur theorem

In mathematics, the Banach–Mazur theorem is a theorem of functional analysis. Very roughly, it states that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur.

Statement of the theorem

Every real, separable Banach space $(X, ||\cdot||)$ is isometrically isomorphic to a closed subspace of $C 0 ([0, 1], R)$ , 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 $C 0 ([0, 1], R)$ 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 $C 0 ([0, 1], R)$ , 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 $x \in X$ , the continuous function $j (x)$ on $K$ is defined by

\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 $C 0 ([0,1] α, R)$ , 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 $i : X \to C 0 [0, 1]$ can be chosen so that every non-zero function in the image $i (X)$ is nowhere differentiable. Put another way, if $D \subset C 0 [0, 1]$ consists of functions that are differentiable at at least one point of $[0, 1]$ , then $i$ can be chosen so that $i (X) \cap D = {0}.$ This conclusion applies to the space $C 0 [0, 1]$ itself, hence there exists a linear map $i : C 0 [0, 1] \to C 0 [0, 1]$ that is an isometry onto its image, such that image under $i$ of $C 0 [0, 1]$ (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 $C 0 [0, 1]$ .

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: 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. JSTOR 2161889. doi:10.2307/2161889.

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert) LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition
Theorems	Banach–Alaoglu Banach–Mazur Banach–Saks Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point
Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function
Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar

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