Vectors

The Riesz representation theorem says that to each bounded linear functional $\xi$ on a Hilbert space $\mathbf{H}$ there corresponds a vector $g$ in $\mathbf{H}$ such that $\xi (f) = \left\langle f, g \right\rangle$ for all $f$. The statement is "invariant" or "coordinate-free," and therefore, according to current mathematical ethics, it is mandatory that the proof be such. The trouble is that most coordinate-free proofs are so elegant that they conceal what is really going on.

In a real vector space (and hence, in particular, in a complex vector space) the segment joining two distinct vectors $f$ and $g$ is, by definition, the set of all vectors of the form $tf + (1-t)g$, where $0 \leq t \leq 1$. A subset of a real vector space is convex if, for each pair of vectors that it contains, it contains all the vectors of the segment joining them. The unit ball is the set of all vectors $f$ with $\left\| f\right\| \leq 1$.

If $h = tf + (1-t)g$ is a point of the segment joining two distinct vectors $f$ and $g$, and if $0 < t < 1$, then $h$ is called an interior point of that segment. If a point of a convex set does not belong to the interior of any segment in the set, then it is called an extreme point of the set. A closed convex set in a Hilbert space is called strictly convex if all its boundary points are extreme points.

A continuous curve in a Hilbert space $\mathbf{H}$ is a continuous function from the closed unit interval into $\mathbf{H}$; the curve is simple if the function is one-to-one. The chord of the curve $f$ determined by the parameter interval $\left[a, b\right]$ is the vector $f(b) - f(a)$. Two chords, determined by the intervals $\left[a, b\right]$ and $\left[c, d\right]$, are non-overlapping if the intervals $\left[a, b\right]$ and $\left[c, d\right]$ have at most an end-point in common.

A crinkled arc is a simple continuous curve with every two non-overlapping chords orthogonal (see Problem 5). There are three trivial senses in which crinkled arcs are not unique. (1) Translation: fix a vector $f_0$ and replace the arc $f$ by $f + f_0$. Remedy: normalize so that $f(0) = 0$. (2) Scale: fix a positive number $\alpha$ and replace the arc $f$ by $\alpha f$. Remedy: normalize so that $\left\| f(1)\right\| = 1$. (3) Span: fix a Hilbert space $\mathbf{H}_0$ and replace $\mathbf{H}$ (the range space of $f$) by $\mathbf{H} \oplus \mathbf{H}_0$. Remedy: normalize so that the span of the range of $f$ is $\mathbf{H}$. A crinkled arc is called normalized in case all three of these normalizations have been applied to it.

There are two other useful ways in which one crinkled arc can be changed into another. One is reparametrization: fix an increasing homeomorphism $\varphi$ of $\left[0,1\right]$ onto itself and replace $f$ by $f \circ \varphi$. The other is unitary equivalence: fix a unitary operator $U$ on $\mathbf{H}$ and replace $f$ by $Uf$.

A subset of a Hilbert space is total if its span is the entire space.

A subset of a Hilbert space is total if its span is the entire space.

The Hilbert space $l^2$ consists, by definition, of all infinite sequences $\left\langle \xi_0, \xi_1, \xi_2, \ldots \right\rangle$ of complex numbers such that $\sum_{n=0}^{\infty } \left|\xi_n\right|^2 < \infty$. The vector operations are coordinatewise and the inner product is defined by

A subset of a Hilbert space is total if its span is the entire space.