Although 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 in 1954 by none other than Alexander Grothendieck (Fields Medal 1966), which in turn had originated within the framework of the theory of distributions of Laurent Schwartz (Fields Medal 1950 and 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.

In 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.

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.

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 the 1986-87 course 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."

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 around 150 pages) 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.

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 about 1950, 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.