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 ${\displaystyle f\colon \left[0,1\right]\longrightarrow X}$, 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 ${\displaystyle 0\leqslant a<b\leqslant c<d\leqslant 1<}$ 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 ${\displaystyle \left[a,b\right]}$ and ${\displaystyle \left[c,d\right]}$ are nonoverlapping.

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,

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

where ${\displaystyle \left\{x_{n}\right\}}$ is an orthonormal set.