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Hilbert, David

theory mathematics algebraic space

(1862–1943) German mathematician: originated the concept of Hilbert space.

Hilbert was educated at the universities of Königsberg and Heidelberg, spending short periods also in Paris and Leipzig. After 6 years as a Privatdozent (unsalaried lecturer) at Königsberg he became a professor there in 1892. In 1895 he was given the prestigious chair in mathematics at Göttingen, which he retained until 1930. He was a talented, lucid teacher and the university became a major focus of mathematical research. Hilbert contributed to analysis, topology, geometry, philosophy and mathematical physics and became recognized as one of the greatest mathematicians in history.

His earliest research was on algebraic invariants, and he both created a general theory and completed it by solving the central problems. This work led to a new and fruitful approach to algebraic number theory which was the subject of his masterly book Der Zahlbericht (trans The Theory of Algebraic Number Fields, 1897). He gathered and reorganized number theory and included many new and fundamental results; this became the basis for the later development of class-field theory.

Abandoning number theory while many problems remained, Hilbert wrote another classic, Grundlagen der Geometrie (Foundations of Geometry), in 1899. It contains fewer innovations but describes the geometry of the 19th-c, using algebra to build a system of abstract but rigorous axiomatic principles. Later, Hilbert developed work on logic and consistency proofs from this. Most important of all, he developed within topology (using his theory of invariants) the concept of an infinite-dimensional space where distance is preserved by making the sum of squares of co-ordinates a convergent series. This is now called Hilbert space, and is much used in pure mathematics and in classical and quantum field theory. His ideas on operators in Hilbert space prepared the way .

Hilbert’s work also gave rise to the ‘Hilbert programme’ of building mathematics axiomatically and using algebraic models rather than intuition. While a productive controversy arose, greatly influencing mathematical philosophy and logic, this formalistic approach was later displaced by work. Hilbert’s views on proof theory were later developed by G Gentzen. In 1900 Hilbert proposed 23 unsolved problems to the International Congress of Mathematicians in Paris. The mathematics created in the solution of many of these problems has shown Hilbert’s profound insight into the subject.

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