He married Gillian Patrick, in 1947. He taught at the University of Edinburgh. He studied at Oklahoma State University from 1950-51. He taught at Newcastle University, with Werner Wolfgang Rogosinski. He was visiting professor at Yale University. He taught at the University of Edinburgh, from 1963 to 1984. In 1966, he was awarded the London Mathematical Society's Berwick Prize. Despite not himself having a PhD, Bonsall supervised many PhD candidates who knew him affectionately as "FFB". After his retirement, Bonsall and his wife moved to Harrogate.
- Compact linear operators, Yale University, Dept. of Mathematics, 1967
- Frank F. Bonsall, John Duncan, Numerical ranges of operators and normed spaces and of all elements of normed algebras, London Mathematical Society
- Frank F. Bonsall, J. Duncan, Numerical ranges II, Cambridge univ. press, 1973
- Alastair Gillespie (4 April 2011). "Professor Frank Bonsall: Leading mathematician of the post-war years who led research into functional analysis". The Independent.
- The Mathematics Genealogy Project
- Royal Society of Edinburgh obituary