History

F. Riesz proves the Riesz Representation Theorem (1911) — click image for original.

In what follows, the functions of bounded variation will play a leading role. We know the importance of this class of functions defined by M. Jordan, whose most remarkable properties become almost obvious after only one statement: that every real function of bounded variation is the difference of two bounded, never decreasing functions.

F. Riesz announces the Riesz Representation Theorem (1909) — click image for original.

To define what is meant by a linear operation, it is necessary to specify the domain of the functional. We consider the totality of all real continuous functions \( \Omega \) between two fixed numbers, for example between \( 0 \) and \( 1 \); for this class, we define the limit function based on the assumption of uniform convergence. The functional operation \( \text{A}[f(x)] \), which associates to each element of \( \Omega \) a corresponding real number, will be called continuous if when \( f(x) \) is the limit of \( f_i(x) \), then \( \text{A}(f_i) \) tends to \( \text{A}(f) \). Such a distributive and continuous operation is said to be linear. It is easy to show that this operation is bounded, that is to say, there is a constant \( M_A \) such that for every element \( f(x) \) we have

\[ \begin{equation}{|\text{A}[f(x)]| \leq M_A \times max. |f(x)|.} \tag{1} \end{equation} \]

Sergei Bernstein.

In 1912 Sergei Bernstein introduced his famous polynomials to prove the Weierstrass Approximation theorem:

If \( F(x) \) is any continuous function in the interval [0,1], it is always possible, regardless how small \( \varepsilon \), to determine a polynomial \( E_n(x) = {a_0 x^n + a_1 x^{n-1} + \cdots + a_n} \) of degree \( n \) high enough such that we have \[ {|F(x) - E_n(x)|} < \varepsilon \] for every point in the interval under consideration.

Weierstrass proved the theorem originally in 1885^[1], the very man who had earlier shown how wild a continuous function can be and in particular, how far from being smooth and subject to a Taylor expansion. Bernstein's proof was simple and based on probability theory. Maven Philip J. Davis says that "while [Bernstein's proof] is not the simplest conceptually, it is easily the most elegant".^[2]

Bernstein on the Weierstrass Approximation Theorem (1912) — click image for original.

I propose to give a very simple proof of the following theorem of Weierstrass:

If \( F(x) \) is any continuous function in the interval [0,1], it is always possible, regardless how small \( \varepsilon \), to determine a polynomial \( E_n(x) = {a_0 x^n + a_1 x^{n-1} + \cdots + a_n} \) of degree \( n \) high enough such that we have \[ {|F(x) - E_n(x)|} < \varepsilon \] for every point in the interval under consideration.

Apostol's ice-cream-cone proof — click image for the entire page.

In high school, the fact that the conic sections are derived from the cone was mentioned in passing, but they were defined in the plane by their equations and tied to their focal properties. So seeing the Dandelin spheres in Apostol fifty years ago was a revelation, effective and surpassingly elegant. He called it the ice-cream-cone proof, virtually a proof by picture that a cone cut obliquely by a plane results in an ellipse as defined by its focal property. The book in question is Calculus, Volume I, by Tom Apostol^[1], among my top three favorite books all-time, first for math. Originally published by Blaisdell in 1961 in large format — unusual in that day, 10" x 7" — Apostol did not condescend, but did not batter you either. After many miles, you look back on an old teacher and think, he did that perfectly, not fully comprehending yourself that he taught you once and for all what mathematical taste is one day in 1965. The book is deeply informed by history altogether apart from the historical introductions, economical and suggestive of hidden depths as they are. Some subjects are one with their own history, philosophy for example, and math is a little like that.

Paul Erdős ~1993.

Bertrand's Conjecture is:

For every integer \( n > 1 \), there is a prime number \( p \) such that:\[ n < p < 2n. \]

It's simple and easy to understand and seems certain, considering the plethora of primes (an infinite number!) and how dense they are — about 6% of all integers in the vicinity of 10,000,000, for example, and reasonably distributed. All the same, it took real work to prove it. Bertrand made the conjecture offhandedly in a celebrated paper in 1845 based on examining numbers up to 6,000,000.^[1] Chebyshev proved it convincingly in 1852 and in a way foreshadowing further developments — this is where he introduced \( \theta(x) = {\sum_{p \leq x} \log p} \)^[2]; Chebyshev's proof was inspired but involved some forbidding calculations leading to the thought, "Surely this can be done easier". It can, as Paul Erdős proved in 1932 in his first published article at the age of 19^[3].

Think of a square in the plane and how it can be rotated around its center to coincide with its original position. There are four rotations altogether — 90°, 180°, 270°, and 360° clockwise, the last bringing the square back to its original configuration. You wouldn't even know the square had been moved unless the corners were somehow distinguished. Starting at the upper left, number the corners 1, 2, 3, 4 in clockwise fashion in order to track the rotations, so that a 90° rotation is identified with the cyclic permutation \( (1 2 3 4) \). In essence, you're rotating around a z axis perpendicular to the plane through the center of the square. You can also rotate around an x, or horizontal, axis through the center of the square. The square comes out of the plane, but is pinned at the middles of the left and right edges as it rotates around that axis through space by 180° — the result is the same whichever way the rotation proceeds, the permutation \( (1 4) (2 3) \). There is a similar vertical rotation, and rotations around each diagonal. These eight rotations form a group, the rigid motions of the square, and the same can be done for any regular polygon. These are the dihedral groups, \( D_4 \) in the case of the square. \( D_n \) has \( 2n \) elements and these groups are nice concrete examples of finite groups.

State Journal front page — Sep 4, 1952.

Scott Milfred is out the other day with Newspaper has stood for public good — for 175 years, meaning the paper he is the editorial page editor of, the Wisconsin State Journal. These folks seem to realize they are almost down for the count and are scurrying around trying to justify themselves and their often baleful enterprise before the final round. Milfred has a taste for authoritarianism (centrist authoritarianism though, no worries), so perhaps it's no surprise to see him taking liberties on the subject of Joe McCarthy and the State Journal — not a 100% supporter!

It is almost beyond belief reading of the antics of the late Senator from Wisconsin, how he cowed the press and political elite, becoming virtual co-president of the republic. Criticize Ted Cruz all you want, he is a statesman compared to McCarthy, a drunken buffoon and known as such by anyone who had ever been in the same room with him. I recommend Richard Rovere's timeless account^[1], written shortly after the fact. Rovere calls McCarthy a gifted demagogue in the first sentence and follows with "no bolder seditionist ever moved among us — nor any politician with a surer, swifter access to the dark places of the American mind."

A. L. Cauchy 1789 - 1857

The history of any subject is like a great river, the central channel meandering but ultimately driving down to the sea, with side branches veering off here and there along the shore to make their own way. Some of those rivulets find their way back to the main course, others peter out in a swamp, the two exchanging places as blockages and torrential rains come into play. Powerful underground springs interact with topography and gravity to govern the web of flowing water in mysterious ways. And all changing year by year as Mark Twain said of the old Mississippi.

Cauchy's early work on permutations was pivotal in the literal sense of that word. He harkened back to the preoccupations and problems of predecessors immediate and remote, recasting them into a new framework whose foundations last to this day and constitute the essential basis of group theory. Granted, everything in sight is a permutation group and it took some time for an abstract group theory to form and emancipate itself from early origins, even to outgrow the connection with algebraic functions and the term substitution (Cauchy's term for permutation). Camille Jordan's path-breaking book in 1870, for example, was titled Traité des substitutions et des équations algébriques, Eugen Netto's in 1882 Substitutionentheorie und ihre Anwendung auf die Algebra.

MM. Lagrange and Vandermonde are, I believe, the first who considered functions of several variables relative to the number of values they can return, when these variables are substituted for each other.