Errett Bishop
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Errett Albert Bishop (1928-1983) was an American mathematician known for is work on analysis. He is the father of constructivist analysis, by virtue of his 1967 Foundations of constructive analysis, where he proved most of the important theorems in real analysis by constructive methods.
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Life
Errett Bishop's father, Albert T Bishop, graduated from the United States Military Academy at West Point, ending his career as professor of mathematics at Wichita State University in Kansas. Although he died when Errett was only 5 years old, he influenced Errett's eventual career by the math texts he left behind, which is how Errett discovered mathematics. Errett grew up in Newton, Kansas. He and his sister were apparent math prodigies.
Bishop entered the University of Chicago in 1944, obtaining both the BS and MS in 1947. The doctoral studies he began in that year were interrupted by two years in the US Army, 1950-52, doing mathematical research at the National Bureau of Standards. He completed his Ph.D. in 1954 under Paul Halmos; his thesis was titled Spectral Theory for Operations on Banach Spaces.
Bishop taught at the University of California, 1954-65. He spent the 1964-65 academic year at the Miller Institute for Basic Research in Berkeley. From 1965 until his death, he was professor at the University of California at San Diego.
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 Cn, and a new proof of Remmert's proper mapping theorem.
- Constructive mathematics. Bishop become 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.
See also
Reference
- Rosenblatt, M., ed., 1985. Errett Bishop: Reflections on him and his research. Proceedings of the memorial meeting for Errett Bishop held at the University of California-San Diego, September 24, 1983. Contemporary Mathematics 39. AMS.
Quotation
- "(A) Mathematics is common sense;
- (B) Do not ask whether a statement is true until you know what it means;
- (C) A proof is any completely convincing argument;
- (D) Meaningful distinctions deserve to be preserved. " (from his 1973 Schizophrenia in contemporary mathematics, reprinted in Rosenblatt 1985)