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 ${\displaystyle f:[0,1]\to X,}$ where ${\displaystyle X}$ is a Hilbert space with inner product ${\displaystyle \langle \cdot ,\cdot \rangle .}$ We say that ${\displaystyle f}$ is a crinkled arc if it is continuous and possesses the crinkly property: if ${\displaystyle 0\leq a<b\leq c<d\leq 1}$ then ${\displaystyle \langle f(b)-f(a),f(d)-f(c)\rangle =0,}$ that is, the chords ${\displaystyle f(b)-f(a)}$ and ${\displaystyle f(d)-f(c)}$ are orthogonal whenever the intervals ${\displaystyle [a,b]}$ and ${\displaystyle [c,d]}$ 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 ${\displaystyle f(t)}$ is a crinkled arc if and only if, after appropriate scaling,

$${\displaystyle f(t)={\sqrt {2}}\,\sum _{n=1}^{\infty }x_{n}{\frac {\sin(n-1/2)\pi t}{(n-1/2)\pi }}}$$

${\displaystyle \left(x_{n}\right)_{n}}$

See also[]

• Infinite–dimensional vector function#Crinkled arcs

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, Springer-Verlag, doi:10.1007/978-1-4615-9976-0
• 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