**Manuel Valdivia**grew up in Martos, a city in south Spain which was very prosperous due to its leading production of olive oil. As he grew up he loved mathematics but had an even greater passion for philosophy, literature and especially for poetry. While studying at the college in his home town, Valdivia read Charles Hermite's proof that e, the base of natural logarithms, is transcendental. He read, at about the same time, the collection of poems

*Sombra del Paraiso*Ⓣ by Vicente Aleixandre. He loved both and that left him in a complete quandary as to whether to study poetry or mathematics. He sat the State examination at the University of Granada and was awarded a Bachelor's degree.

Perhaps his next move was one of the most surprising things he did, for he now enrolled for a law degree in the Faculty of Law of Madrid University. Since his passions were mathematics and poetry, it is not hard to understand that the law course went badly for Valdivia. He gave up and decided to enter the School of Agronomist Engineers at Polytechnic University of Madrid. This was not quite so strange a move as the law degree, for he realised that, with the mathematical skills he already had, he was able to solve problems in the entry examination which others who had spent years preparing to take this examination were unable to solve. This time he successfully completed the degree course and graduated with an Agronomist Engineering degree in 1958. He was awarded a scholarship to undertake research in agronomy and was appointed as an adjunct professor of mathematics in the School of Agronomist Engineers at Polytechnic University of Madrid. He now undertook two studies in parallel. He undertook research for a doctorate in Agronomist Engineering and was awarded this degree in 1961. At the same time he studied as an undergraduate for a degree in mathematical science at the Complutense University of Madrid and was awarded the degree also in 1961.

Valdivia met Dario Maravall Casesnoves (1923-2016) who had been awarded a doctorate in mathematical sciences from the Complutense University of Madrid in 1956 and a doctorate in agronomist engineering from the School of Agronomist Engineers at the Polytechnic University of Madrid in 1958. Dario Maravall helped Valdivia continue his move towards pure mathematics by introducing him to Ricardo San Juan Llosá who had studied for his doctorate under Julio Rey Pastor. Ricardo San Juan became his thesis advisor and quickly realised that Valdivia had extraordinary talents for mathematics. Valdivia was awarded his doctorate in 1963 for his thesis *Sucesiones de aplicaciones continuas, absolutamente continuas y sumables con diversos tipos de convergencia* Ⓣ. This thesis was devoted to different concepts of convergence of sequences of functions in a metric space.

Following the award of his mathematics doctorate, Valdivia was awarded a fellowship by the "Juan March" Foundation for Research in Mathematics in 1963. Two years later he competed for the position of Professor of Mathematical Analysis II and III in the Faculty of Sciences of the University of Valencia and was successful. In the same year of 1965 he published two papers: *Algunos criterios de convergencia en la teoria de la integración* Ⓣ and *Desarrollos asintóticos y familias compactas de funciones holomorfas * Ⓣ. In 1967 his research interests changed after reading the work of Gottfried Köthe and Helmut H Schaefer on topological vector spaces. This was clear from his next two publications which appeared in 1968: *El teorema general de la gráfica cerrada en los espacios vectoriales topológicos localmente convexos* Ⓣ and *El teorema general de la applicación abierta en los espacios vectoriales topológicos localmente convexos* Ⓣ.

In 1969 Valdivia was appointed to the Chair of Algebra and Infinitesimal Calculus at the Escuela Técnica Superior de Ingenieros Agrónomos of the Polytechnic University of Valencia. An overview of his contributions are given in [12]:-

For example, his papers on weak compactness includeAlthough Valdivia's academic activity has developed in Valencia, his influence has been considerable in the landscape of mathematical analysis in Spain, and especially in the establishment of functional analysis as one of the main research areas. The pioneering aspect of Valdivia's mathematical research must also be emphasized, since when he began research in the early sixties mathematical research in Spain was practically nonexistent, at least according to international standards, which he immediately attained with multiple functional analysis results. For example, as an eloquent illustration, he solved some open issues raised in1954by none other than Alexander Grothendieck(Fields Medal1966), which in turn had originated within the framework of the theory of distributions of Laurent Schwartz(Fields Medal1950and director of Grothendieck's doctoral thesis). Investigations by Valdivia have focused on a surprising variety of topics: analysis and topology, geometry of Banach spaces, Fréchet spaces and locally convex spaces, compact spaces, real analytic functions, Schwartz distributions, polynomials, multilinear forms, ... . We highlight, also as an illustration, his interest in weak compactness in Banach spaces, the study of which led him to define the notion of compact space that has been fertile for many researchers and is now known as the compact space of Valdivia.

*Some criteria for weak compactness*(1972),

*On weak compactness*(1973) and

*Some new results on weak compactness*(1977). The last mentioned paper begins:-

Three problems posed by Alexander Grothendieck were solved by Valdivia in the papersIn this paper we prove that if E is a quasi-complete locally convex space and A is a subset of E such that every weakly continuous real-valued function on E is bounded in A, then A is weakly relatively compact. As consequence, a Banach space E is reflexive if and only if every weakly continuous real valued function on E is bounded in the closed unit ball.

*Solution of a problem of Grothendieck*(1979),

*On quasi-normable echelon spaces*(1981) and

*A characterization of totally reflexive Fréchet spaces*(1989).

In the citation delivered by Manuel López Pellicer on the occasion that Manuel Valdivia was given an Honorary Doctorate by the Polytechnic University of Valencia on 27 September 1993, his qualities as a teacher were described [11]:-

On that same occasion López Pellicer also described aspects of Valdivia's character [11]:-In addition to being an excellent researcher, Valdivia is a magnificent teacher of great clarity who always conveys enthusiasm in all the subjects he explains, uniting a great depth and a very varied knowledge of the different branches of mathematics. It was said of the French mathematician Hermite that those who had had the joy of being his pupils could not forget the feeling of beauty and clarity that he made run through his audience. These words can be applied to Professor Valdivia. And that's why he has put in enormous work to train teachers and researchers, as well as his support to other researchers, both in the two Universities of Valencia, as well as in other Spanish and foreign universities. He has directed more than thirty doctoral theses and there are many articles with thank-you notes to Professor Valdivia.

For a version of Valdivia's reply on the occasion of being given an Honorary Doctorate by the Polytechnic University of Valencia, see THIS LINK.Regarding the human aspect of this great mathematician in his love for philosophy, poetry and music, I will try to describe some facet through ideas he has written or I have heard. Some years ago he told me that what he asked of life when he was old was to be able to continue doing mathematics. In another way he returned to describing this idea in the speech he gave at the opening lecture of the1986-87course at the University of Valencia when he told us that he felt he identified with the philosophical spirit of the Latin poet Virgil when faced with the attempt of his friends to dissuade him from studying Greek in the last years of his life responded bluntly: "You have to work, you have to study as if you are never going to die." In that same lecture another facet of his spirit could be seen when he stated that in difficult moments, or when the means to work that were within his reach were meagre, he had never allowed the sadness of prevent him using all possibilities, quoting the phrase of the Bengali poet Rabindranath Tagore: "If you cry at night because you can not see the sun, tears will prevent you from looking at the stars."

Valdivia was also awarded an honorary doctorate by Alicante University on 2 May 2000. In reply he made the speech "Aesthetic and intellectual aspects of mathematics", a version of which we give at THIS LINK.

Valdivia was also awarded an honorary doctorate by the Jaume I University in Castelló de la Plana, Spain, in 2001. In reply he delivered the address "Infinity in mathematics", a version of which we give at THIS LINK.

In 1982 Valdivia published the 510-page monograph *Topics in locally convex spaces*. Nigel J Kalton wrote in a review:-

We have already mentioned some honorary degrees that Valdivia was awarded. Let us list here the main honours which were bestowed on him: elected to the Real Academia de Ciencias Exactas, Fisicas y Naturales (1975); awarded the Great Cross of Alfonso X the Wise for Teaching Merit (1981); elected to the Academia de Ciencias y Artes de Barcelona (1985); elected to the Société Royale des Sciences de Liège (1990); awarded an honorary doctorate by the Universidad Politécnica de Valencia (1993); awarded an honorary doctorate by the University Jaume I (1993); awarded an honorary doctorate by the University of Liège (1994); elected to the Real Academia de Ingenieria (1995); elected to the Academia Canaria de las Ciencias (1999); Confederación Espanola de Organizaciones Empresariales prize (1999); awarded an honorary doctorate by the Universidad de Alicante (2000); awarded an honorary doctorate by the Universidad de Jaén (2001); awarded the first gold medal of the Consell Valencià de Cultura (2001); elected an honorary member of the Real Academia de Medicina i Ciències Afins de la Comunitat Valenciana (2003); appointed Honorary Professor of the University of Valencia (2008).This is a study of certain aspects of current developments in the area of locally convex linear topological spaces by one of the most active researchers in the field. The selection of topics, is, as the author admits, heavily influenced by his research interests. There are three chapters(each of around150pages)which are further subdivided into sections. In Chapter I, many different types of locally convex spaces are introduced and studied. ... The second chapter concerns sequence spaces. ... The third chapter is concerned with function spaces. ... The selection of material is clearly guided throughout by the author's own research. Occasionally it is difficult to see the underlying theme because of the way in which the material has been selected. It is clearly the author's intention to present in book form those aspects of the theory of locally convex spaces which interest him, without attempting a coherent overview. There is a wealth of information here, some of which is not readily accessible anywhere else. It will obviously be extremely valuable to anyone working in this area.

From 3 to 7 July 2000 the Functional Analysis Valencia 2000 conference was held in Valencia. This International Functional Analysis Meeting in Valencia, held to celebrate the 70th Birthday of Manuel Valdivia, was a Satellite Conference to the Third European Congress of Mathematics held in Barcelona. Organized jointly by the Polytechnic University of Valencia and the University of Valencia, it had organizers from the Universities of Kent (USA), Paderborn (Germany), Barcelona (Spain), Zürich (Switzerland) and Liège (Belgium). With more than 300 attendees and 250 lectures and presentations, it was the largest functional analysis conference held in the year 2000, the World Year of Mathematics.

Let us end this biography by quoting the following summary of Valdivia's contributions [5]:-

Manuel Valdivia was an excellent mathematician, a great scientist and a deep intellectual, who loved literature and especially poetry. He conceived his mathematical creation, for which he had an extraordinary talent, as something similar to artistic creation and appreciated both beauty in mathematics and art. He worked in a wide variety of subjects, always with a passion and perseverance, which he knew how to convey to his students. He made essential contributions to many areas of functional analysis, solved very difficult problems proposed, inter alia, by Schwartz and Grothendieck about1950, which had remained unresolved for many years. He obtained many profound results, opened new directions of investigation and always opted for quality in his research. His numerous works were published in the best mathematics journals, had a great impact and influence and received many citations. The great success of mathematical research and functional analysis in Spain would be inconceivable without the pioneering work of Manuel Valdivia.

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