Bonsall was born on 31 March 1920, the youngest son of Wilfred C Bonsall and Sarah Frank. His older brother was Arthur Bonsall. He graduated from Bishop's Stortford College in 1938, and studied at Merton College, Oxford. He served in World War II, in the Corps of Royal Engineers, and in India from 1944 to 1946.
He married Gillian Patrick, in 1947. He lectured at the University of Edinburgh from 1947-1948; was Visiting Associate Professor at Oklahoma State University from 1950-51; taught at Newcastle University, with Werner Wolfgang Rogosinski in the 1950s; was visiting professor at Yale University; and 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, 1971
- Frank F. Bonsall, J. Duncan, Numerical ranges II, Cambridge univ. press, 1973
- Frank F. Bonsall, J. Duncan, Complete normed algebras, 1973
- Levens, R.G.C., ed. (1964). Merton College Register 1900-1964. Oxford: Basil Blackwell. p. 290.
- 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
- Rickart, C. E. (1975). "Review: Complete normed algebras by F. F. Bonsall and J. Duncan" (PDF). Bull. Amer. Math. Soc. 81, Part 1 (3): 514–522. doi:10.1090/s0002-9904-1975-13727-x.