Work
Bishop's wide-ranging work falls into five categories:
- Polynomial and rational approximation. Examples are extensions of Mergelyan's approximation theorem and the theorem of Frigyes Riesz and Marcel Riesz concerning measures on the unit circle orthogonal to polynomials.
- The general theory of function algebras. Here Bishop worked on uniform algebras (commutative Banach algebras with unit whose norms are the spectral norms) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures. Bishop wrote a 1965 survey "Uniform algebras," examining the interaction between the theory of uniform algebras and that of several complex variables.
- Banach spaces and operator theory, the subject of his thesis. He introduced what is now called the Bishop condition, useful in the theory of decomposable operators.
- The theory of functions of several complex variables. An example is his 1962 "Analyticity in certain Banach spaces." He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in, and a new proof of Remmert's proper mapping theorem.
- Constructive mathematics. Bishop became interested in foundational issues while at the Miller Institute. His now-famous Foundations of Constructive Analysis (1967) aimed to show that a constructive treatment of analysis is feasible, something about which Weyl had been pessimistic. A 1985 revision, called Constructive Analysis, was completed with the assistance of Douglas Bridges.
In 1972, Bishop (with Henry Cheng) published Constructive Measure Theory. In the later part of his life Bishop was seen as the leading mathematician in the area of Constructive mathematics. In 1966 he was invited to speak at the International congress of mathematics on constructive mathematics. His talk was titled "The Constructivisation of Abstract Analysis." The American mathematical society invited him to give four hour long lectures as part of the Colloquium Lectures series. The title of his lectures was "Schizophrenia of Contemporary Mathematics." A. Robinson wrote of his work in constructive mathematics: "Even those who are not willing to accept Bishop's basic philosophy must be impressed with the great analytical power displayed in his work." (Warschawski 1985) Robinson wrote in his review of Bishop's book that Bishop's historical commentary is "more vigorous than accurate".
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