Crinkled arc

In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos.

Specifically, consider $f\colon [0,1]\to X,$ where $X$ is a Hilbert space with inner product $\langle \cdot ,\cdot \rangle .$ We say that $f$ is a crinkled arc if it is continuous and possesses the crinkly property: if $0\leq a<b\leq c<d\leq 1$ then $\langle f(b)-f(a),f(d)-f(c)\rangle =0,$ that is, the chords $f(b)-f(a)$ and $f(d)-f(c)$ are orthogonal whenever the intervals $[a,b]$ and $[c,d]$ are non-overlapping.

Halmos points out that if two nonoverlapping chords are orthogonal, then "the curve makes a right-angle turn during the passage between the chords' farthest end-points" and observes that such a curve would "seem to be making a sudden right angle turn at each point" which would justify the choice of terminology. Halmos deduces that such a curve could not have a tangent at any point, and uses the concept to justify his statement that an infinite-dimensional Hilbert space is "even roomier than it looks".

Writing in 1975, Richard Vitale considers Halmos's empirical observation that every attempt to construct a crinkled arc results in essentially the same solution and proves that $f(t)$ is a crinkled arc if and only if, after appropriate normalizations, $f(t)={\sqrt {2}}\,\sum _{n=1}^{\infty }x_{n}{\frac {\sin(n-1/2)\pi t}{(n-1/2)\pi }}$ where $\left(x_{n}\right)_{n}$ is an orthonormal set. The normalizations that need to be allowed are the following: a) Replace the Hilbert space H by its smallest closed subspace containing all the values of the crinkled arc; b) uniform scalings; c) translations; d) reparametrizations. Now use these normalizations to define an equivalence relation on crinkled arcs if any two of them become identical after any sequence of such normalizations. Then there is just one equivalence class, and Vitale's formula describes a canonical example.

References

Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781.
Halmos, Paul R. (1982), A Hilbert Space Problem Book, Graduate Texts in Mathematics, vol. 19, Springer-Verlag, doi:10.1007/978-1-4615-9976-0, ISBN 978-1-4615-9978-4
Vitale, Richard A. (1975), "Representation of a crinkled arc", Proceedings of the American Mathematical Society, 52: 303–304, doi:10.1090/S0002-9939-1975-0388056-1

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

See also

References