Bücher der Reihe Grundlehren der mathematischen Wissenschaften

It is the author's aim to give a systematic account of the most im portant ideas, methods and results of the theory of topological vector spaces.

In preparing the Bibliography, works having an English equivalent have been translated into the English title, but in the text the referenee to the Russian work was retained, as it was impraetieal to attempt to find in eaeh ease the eorresponding eitation in the English edition.

Translation of Controle optimal de systemes gouvernes par des equations aux derivees partielles.

Besides the possibility of the mutual fruitful interaction between celestial and oscillatory mechanics which is thereby created, our linear differential equations are as a result everywhere regular.

By "non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be given in function space s F and G , F being a space" on m" and the G/ s spaces" on am" ; j we seek u in a function space u/t "on m" satisfying (1) Pu = f in m, (2) Qju = gj on am, 0 ~ i ~ 'v"])).

the points at which the continuity of the solution of the Dirichlet problem may be violated) was completely solved and a procedure to obtain a generalized solution to the Dirichlet problem was described.

In Chapter 6, the results of Chapter'> 4 and 5 are applied to the study of optimal control problems for systems governed by evolution equations, when the control appears in the boundary conditions (so that non-homogeneous boundary value problems are the basic tool of this theory).

The purpose of this book is to give a unified treatment of the limit theory of branching processes. There is a natural classification of branching processes according to their criticality condition, their time parameter, the single or multi-type particle cases, the Markovian or non-Markovian character of the pro cess, etc.

The theory of harmonic spaces, sometimes also called axiomatic theory of harmonic functions, plays a particular role among the above mentioned theories.

Topological Dynamics has its roots deep in the theory of differential equations, specifically in that portion called the "qualitative theory".

1) - the function cp is continuous, vanishes for v = 0 and satisfies the inequality cp( v)v > 0 for every v =/= o. The object of the study of absolute stability consists in finding a criterion which secures simultaneously the stability of all the systems characterized by Condition (6) .

This book is a fully detailed introduction to the theory of modular functions of a single variable. An extensive discussion of the modular group SL(2, Z) is followed by the introduction to the theory of automorphic functions and auto morphic forms of integral dimensions belonging to SL(2,Z).

In the remainder of the book we treat a closely related topic: Loewner's theory of monotone matrix functions and his analytic continuation theorem which guarantees that a real function on an interval of the real axis which is a monotone matrix function of arbitrarily high order is the restriction to that interval of a Pick function.

The purpose of this book is to present the main structure theorems in the isometric theory of classical Banach spaces. A classical Banach space is a Banach space X whose dual space is linearly isometric to Lp(j1, IR) (or Lp(j1, CC) in the complex case) for some measure j1 and some 1 ~ p ~ 00.

An ultrafilter is a truth-value assignment to the family of subsets of a set, and a method of convergence to infinity. The model-theoretic property finds its expression in the construction of the ultraproduct and the compactness type of theorem of Los (implying the compactness theorem of first-order logic);

On the other hand, many problems of non linear analysis are still far from a solution (problems arising from the internal development of mathematics and, in particular, problems arising in the process of interpreting new problems in the natural sciences).

the details in Chapter 1, Section 2.2.1) that au (aZu azu aZu) (1) in Q, t>o, -a - du = g du = -a z + -a z + -a z t Xl X X3 z l g a given function, with boundary conditions in the form of inequalities u(X,t"o => au(x,t)/an=O, XEr, (2) u(x,t)=o => au(x,t)/an?:O, XEr, to which is added the initial condition (3) u(x,O)=uo(x).

The translator of a mathematical work faces a task that is at once fascinating and frustrating. The translator records his gratitude to Linda Sax, who typed the entire translation, to Laura Larsson, who prepared the bibliography (considerably modified from the original), and to Betty Underhill, who rendered essential assistance.

The works of Jaak Peetre constitute the main body of this treatise. Peetre suggested to the second author, J. Lofstrom, writing a book on interpolation theory and he most generously put at Lofstrom's disposal an unfinished manu script, covering parts of Chapter 1-3 and 5 of this book.

TO THE SECOND EDITION Since publication of the First Edition several excellent treatments of advanced topics in analysis have appeared.

The numerous explicit formulae of the classical theory of quadratic forms revealed remarkable multiplicative properties of the numbers of integral representations of integers by positive definite integral quadratic forms.