**Frank Bonsall**'s parents were Sarah and Wilfred Cook Bonsall. Sarah was from the North of England, having been brought up in North Yorkshire, while Wilfred was a Derbyshire man. Sarah and Wilfred had two children: the elder was Arthur Wilfred Bonsall and the younger Frank, the subject of this biography. Let us note at this stage that Arthur Bonsall went on to become the director of the Government Communications Headquarters at Cheltenham from 1973 to 1978 and was knighted in 1977.

When Frank was three years old, in 1923, the family moved to Welwyn Garden City and he was brought up in this new town. The town, about 30 km north of London, was only founded in 1920 and was a small place when the Bonsalls moved there. Frank was educated at Fretherne House Preparatory School in Welwyn Garden City until he was thirteen years old when his parents sent him to Bishops Stortford College as a boarder in 1933. This independent school, for both day pupils and boarders, took only boys. Despite not being particularly happy at the boarding school, Bonsall excelled academically enjoying particular success in mathematics. In 1938 he graduated from Bishops Stortford College and matriculated at the University of Oxford, entering Merton College, one of the oldest of the Colleges founded in the 13^{th} Century, to study mathematics [2]:-

He did not get the chance to enjoy much more of his university career at this time for World War II broke out in September 1939 when German troops invaded Poland. In 1940 he was called up for military service and he became a cadet in the Corps of Royal Engineers. On 21 June 1941 he was promoted from cadet to 2Later, he recalled how much he enjoyed the freedom of university life, as well as his first encounters with rigorous analysis, the area of mathematics that was to become his speciality.

^{nd}lieutenant after traing at the Officer Cadet Training Unit. Bonsall spent six years in the Royal Engineers, spending two years in 1944-46 in India where his task was to try out equipment to test how it functioned under extreme jungle conditions. He did not completely give up mathematics during these years for he took Edward Titchmarsh's book

*The theory of functions*(1932) with him. It is said that "a generation of mathematicians learned the theory of analytic functions and Lebesgue integration from [Titchmarsh's book], and also learned (by observation) how to write mathematics" and, indeed Bonsall was one of this "Titchmarsh generation".

After the end of World War II in 1945 it took a while before things began to return to normal and for some time Bonsall continued to serve in India. He returned to take up his undergraduate studies at Oxford in 1946 and this proved a significant year in more ways than one since he quickly became friends with Gillian Patrick, one of his fellow mathematics undergraduates at Oxford. Jill, as she was known to her friends, became Bonsall's wife in 1947. After the award of his first degree, Bonsall had to choose between remaining at Oxford to undertake research or take up a temporary one-year lectureship at the University of Edinburgh in Scotland. He chose to spend the year lecturing in Edinburgh and there he met W W Rogosinski who had just been appointed to the chair of mathematics at Newcastle. In the following year, Bonsall moved to Newcastle to join Rogosinski and to take up a lectureship there. Bonsall had had no research training before his first appointment but the influence of Rogosinski guided the young talented lecturer into becoming an excellent researcher. His first paper *Note on a theorem of Hardy and Rogosinski* (1949) contained an elementary proof of a known theorem. His next papers were *On generalized subharmonic functions* (1950), *The characterization of generalized convex functions *(1950) and *Inequalities with non-conjugate parameters* (1951). In this last mentioned paper, R P Boas, Jr explains in a review, that Bonsall:-

Bonsall spent the academic year 1950-51 on study leave on the Stillwater campus of Oklahoma State University. Up to this time his research had been along fairly classical lines but at Stillwater he began to undertake research in functional analysis, the topic on which he undertook research for the rest of his life. In collaboration with Morris Marden, Bonsall published... discusses a variety of inequalities, mostly connected with Hilbert's double series theorem ... . He applies a new technique which yields much simplified proofs of known results as well as new results.

*Zeros of self-inversive polynomials*(1952) and, in collaboration with Alfred Goldie,

*Algebras which represent their linear functionals*(1953). Back in Newcastle after his year at Stillwater, Bonsall soon became one of the leading functional analysts, becoming a leading expert on Banach algebras. When Rogosinski retired from the Newcastle chair in 1959, Bonsall was appointed to succeed him. Alastair Gillespie writes [2]:-

Let us look briefly at some of the books that Bonsall published. His first wasIn1963, the University of Edinburgh instituted a second chair of mathematics, the Maclaurin chair. Bonsall applied and was interviewed by a committee chaired by the principal, Sir Edward Appleton, who immediately wrote to offer him the position. This was one of Appleton's last acts as he died suddenly that night. Bonsall took up the chair in1965but spent the following academic year at Yale, where he was already committed to a visiting professorship. On his return he did much to strengthen the position of functional analysis. Along with John Ringrose and Barry Johnson at Newcastle, he founded the North British Functional Analysis Seminar, one of the first inter-university seminars in mathematics, and a model for many others.

*Lectures on some fixed point theorems of functional analysis*(1962). This book was the lecture notes from a course Bonsall had given at the Tata Institute of Fundamental Research, Bombay, India. H H Schaefer writes in a review:-

Bonsall's other books were all written in collaboration with John Duncan, who had been one of his research students graduating with a Ph.D. from Newcastle in 1964. The first of these books wasThe present set of notes is concerned with the application of several fixed point principles(in particular, the Brouwer-Schauder-Tikhonov theorem)to linear and nonlinear functional analysis. The exposition is easy to read, and though a more accurate reproduction(omission of symbols)would do the underlying lectures greater justice, these notes supply a great deal of information to readers with even a modest mathematical background. But the expert also is likely to find something of interest to him in each of the different sections.

*Numerical ranges of operators on normed spaces and of elements of normed algebras*(1971) to which the authors added a second volume

*Numerical ranges. II*(1973). T W Palmer, reviewing the second of these books, writes:-

The other important text jointly authored by Bonsall and Duncan wasThis book is a continuation of the authors' work 'Numerical ranges of operators on normed spaces and of elements of normed algebras'. The text of the present book was prepared with astonishing rapidity after the July1971, Conference on Numerical Ranges held at Aberdeen University. With the exception of a body of results on semigroup generators, the book together with its predecessor discusses essentially all developments in the subject through1971.

*Complete normed algebras*(1973). I Suciu writes:-

In 1984, Bonsall retired from his chair at the University of Edinburgh and, together with his wife Jill, moved to Harrogate and there he was able to indulge his passion for gardening as well as continuing to undertake research and write papers. He often attended seminars at the University of Leeds and at the University of York, both within easy access of Harrogate. In 2002 Bonsall and his wife moved into a retirement home in Harrogate and there he spent the final years of his life.The authors of this book, as they themselves state in the Introduction, "have devoted almost the whole of our professional lives to the study of Banach algebras". From this large experience they extract a useful and attractive presentation of the principal lines of the general theory of Banach algebras, having constantly in mind the needs of people working in the deep problems which appear either in special Banach algebras like C*-algebras, operator and function algebras, group algebras, etc., or in the spectral theory of operators.

Bonsall received many honours to mark his mathematical contributions. These honours include election as a fellow of the Royal Society of Edinburgh in 1966 and to the Royal Society in 1970. The University of York awarded him an honorary degree in 1990. He gave great service to the London Mathematical Society serving on the Council - the Society awarded him their Senior Berwick Prize in 1966. He served the Edinburgh Mathematical Society as President in 1976-77. He also served on several committees of the Royal Society and on the Science Research Council.

Finally we should mention Bonsall's passion for climbing the Scottish mountains [3]:-

Beyond mathematics, Bonsall had a great interest in mountain climbing, ascending his2801977^{th}Munro in. He also contributed to the debate as to when two close tops count as separate Munros(that is, as separate mountains of height at least3000feet). In two articles in the Scottish Mountaineering Club Journal in1973and1974, Bonsall developed a rule for determining this. His rule yields a list very close to the original one compiled by Sir Hugh Munro in1891and has influenced some subsequent revisions of Scottish Mountaineering Club's definitive list.

**Article by:** *J J O'Connor* and *E F Robertson*

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