David Hilbert (1862-1943) Historical Sketch
David Hilbert was born early in 1862 in a city that is well known among mathematicians, Königsberg, then a part of Germany but now in Russia. To see one reason mathematicians know about Königsberg, go to http://math.youngzones.org/Konigsberg.html.
But another reason that mathematicians know of this city is that so many mathematicians were born there, the most illustrious of whom are Carl Neumann, Alfred Clebsch, Christian Goldbach, Kurt Hensel, Ludwig Otto Hesse, David Hilbert, Gustav Kirchhoff, Rudolf Lipschitz, Johann Rosenhain, Heinrich Schroeter, and Arnold Sommerfeld.
One measure of the importance of Hilbert's place in mathematics is that Carl Boyer, in his beautiful book A History of Mathematics, devotes ten pages to describe his work.
Hilbert attended the University of Königsberg, obtaining his Ph.D. under the direction of Ferdinand von Lindemann (about whom you can read in these Historical Sketches) in 1885. He then taught there until 1895, as a full professor during the last two years. He was subsequently appointed to a chair of mathematics at the University of Göttingen in the German city of that name, where he spent the rest of his career (and life, dying in early 1943). When Hitler came to power in Germany, Hilbert reportedly used his prestige and influence to help younger mathematicians emigrate.
Hilbert produced a set of axioms for Euclidean geometry, and on the basis of this success, he set about in the 1920s to produce a formalistic foundation for classical mathematics, which came to be known as Hilbert's Program, and a Beweistheorie---proof theory---by which all true theorems in mathematics could be established in a finite number of steps. However, the mathematician Kurt Gödel proved in 1931 that such a goal is unattainable; i.e., that in any consistent theory, there are always true theorems that cannot be proved to be true. (This means that mathematicians will always have jobs!)
One of the numerous things that Hilbert did was to propose, at the 1900 International Congress of Mathematicians held in Paris, 23 problems for mathematicians in the 20th century. Their current status can be found in http://en.wikipedia.org/wiki/Hilbert's_problems.
One of the best known of Hilbert's many constructs is what is called Hilbert space. This is an infinite-dimensional space, where the points are infinite sequences a1,a2,a3,...,an,... that have the property of being square summable; i.e., for which the series
converges to some finite number.
a. Show that the harmonic sequence 1, (1/2), (1/3),..., (1/n),... is a point in Hilbert space.
b. Give another example of a point in Hilbert space.
