In 1917 Frank, already well taught by his parents, entered primary school :-
With a mental age far ahead of his actual years, however, he was rather a misfit in school; he did not easily get on with other children, and matters were not helped by frequent, and at times prolonged, absences due to ill health.In 1921 he struck lucky since he was accepted by W H Roberts, a man who loved the challenge of teaching those who did not fit into the usual educational set-up. Roberts was so impressed by Frank that he accepted him free of charge. This proved especially beneficial since, around this time, Frank's father was black-listed by engineering firms because of his trade union activities and so could not get a job. Roberts did an excellent job and ensured that Frank was in a good position to take the entrance examination for Edinburgh University in March 1927. He passed the examination and matriculated at Edinburgh University in October 1927.
With his father unable to obtain work, things were very difficult for Smithies in his first year at Edinburgh University, but it became easier in his second year when he was awarded a Spence Bursary. He graduated in 1931 as the top mathematics student in his year, winning the Napier Medal, the Gadgil Prize, and a John Edward Baxter bursary which paid for two further years study at Cambridge. He had also won a scholarship for St John's College, and he entered the College to study for the Mathematical tripos in October 1931. There he took courses by G H Hardy on Fourier analysis, John Whittaker on integral equations, and Ebenezer Cunningham on mechanics. In fact this was the second member of the Whittaker family who had lectured to Smithies for he had studied the history of mathematics with John Whittaker's father Edmund Whittaker while at Edinburgh University. This is particularly significant since Smithies developed a love of the history of mathematics and made many excellent contributions which we discuss towards the end of this biography.
Smithies graduated in 1933 and began research on integral equations with Hardy at Cambridge. He was influenced, by reading books by Banach and Stone, and attending lectures by Courant and von Neumann, to become interested in functional analysis despite Hardy's dislike of abstract mathematics. He won the Rayleigh Prize in 1935 for an essay on differential equations of fractional order, and was awarded his doctorate for his thesis The Theory Of Linear Integral Equation which he submitted to the University of Cambridge in 1936.
Funded by a Carnegie Fellowship and a St John's College studentship, Smithies then spent two years at the Institute for Advanced Study at Princeton. There he did some joint work with von Neumann and some with R P Boas. One of the notable results of a collaboration with Boas was the article Contribution to the Mathematical Theory of Big Game Hunting. See THIS LINK. After returning to Cambridge in 1938, Smithies taught at St John's College and continued his research. However from the summer of 1940 he was engaged in war work at the Ministry of Supply. His work there was very varied, involving theoretical and experimental work on anti-aircraft guns, statistical work on quality control, and the responsibility for "miscellaneous mathematical problems". In 1942 he helped set up the Advisory Service on Statistical Quality Control in the Ministry of Supply and about the same time he met Nora Arone who had just started working for the Ministry. Smithies took up his duties again at St John's College in September 1945 and three months later he married Nora.
Smithies early work was on integral equations and in 1958 his text Integral equations was published by Cambridge University Press in their Cambridge Tracts in Mathematics and Mathematical Physics Series. He states in the preface that:-
... the present work is intended as a successor to Maxime Bôcher's tract, "An introduction to the study of integral equations" (University Press, Cambridge, 1909).As I A Barnett points out in a review:-
This remark is indeed an understatement, since the present work is much more than a continuation of Bôcher's treatment, good as it was fifty years ago. It must have taken a tremendous amount of reading and sifting of the enormous literature to give a modernized version of the theory in this little volume of scarcely more than 150 pages.Although he published little in the way of original research in functional analysis after 1940, Smithies was the greatest influence on the development of the subject in Britain. This was through his teaching at Cambridge, the large number of research students he guided into the subject, and the lectures he gave around the country. Given this leading role, it is fascinating to read his account in  of how the subject developed, and in particular the role of linear operators in this development. Smithies looks in detail at the development of the concept of an adjoint operator in the years before the Hahn-Banach theorem.
His teaching at Cambridge is discussed in :-
In his teaching at all levels Smithies was consientious and meticulous. ... His lectures, at both undergraduate and postgraduate level, were a source of inspiration - not, perhaps, in the conventional sense as a result of charismatic presentation, but rather because his excellent judgement as to choice of material, combined with exceptional clarity of exposition, allowed the subject matter to speak for itself.After Smithies retired in 1979 he began to take an interest in the history of mathematics, which he had from his undergraduate days, to a more serious level. He undertook some excellent historical research, publishing a number of important works. In fact he expressed a certain sadness that he had not taken up research in this area much earlier in his life. It was work which he simply adored. In fact he had published Weierstrass's theory of the real numbers in 1975 before he retired. It is an interesting account which first shows how Weierstrass constructed the rational numbers from the natural numbers. It then shows that essentially the same technique allowed Weierstrass to construct the real numbers from the rationals. In 1982 Smithies published the paper The background to Cauchy's definition of the integral. This was followed by Cauchy's conception of rigour in analysis in 1986 and finally in 1997 by the masterly work Cauchy and the creation of complex function theory . Smithies last paper was A forgotten paper on the fundamental theorem of algebra published in the Notes and Records of the Royal Society of London in 2000. Let us end this description of Smithies historical work by quoting his own summary of this paper:-
In 1798, there appeared in the Philosophical Transactions of the Royal Society a paper by James Wood, purporting to prove the fundamental theorem of algebra, to the effect that every non-constant polynomial with real coefficients has at least one real or complex zero. Since the first generally accepted proof of this result was given by Gauss in 1799, Wood's paper deserves careful examination. After giving a brief outline of Wood's career, I describe the argument of his paper. His proof turns out to be incomplete as it stands, but it contains an original idea, which was to be used later, in the same context, by von Staudt, Gordan and others, without knowledge of Wood's work. After putting Wood's work in context, I conclude by showing how his idea can be used to prove the complex form of the fundamental theorem of algebra, stating that every non-constant polynomial with complex coefficients has at least one zero in the complex field.Smithies' wife Nora died in 1987 after a long illness. His sister Violet, who had had kept in close contact with throughout his life, died two years later :-
Although physically rather frail during his last few years, Smithies remained mentally alert until the end. He died after a brief illness ...
Article by: J J O'Connor and E F Robertson