Spaces of analytic functions on inductive/projective limits of Hilbert spaces.

*(English)*Zbl 0662.46026
Eindhoven (Netherlands): Technische Univ. Eindhoven,. IV, 170 p. (1988).

This thesis of the author is a study of spaces of analytic functions on sequence spaces. Both the sequence spaces and analytic function spaces are inductive or projective limits of Hilbert spaces or both at once. By means of these analytic function spaces, the concept of “symmetric Fock space” for Hilbert spaces is generalized to sequence spaces and some natural constructions are carried out in this context, such as test spaces and distribution spaces.

The thesis is very well presented and written in four Chapters. Chapter I contains prerequisites. Chapter II is devoted to the Aronszajn’s theory of reproducing kernels and functional Hilbert spaces. Chapter III is an exposition of inductive and projective limits of Hilbert spaces. Chapter IV is devoted to spaces of analytic functions on sequence spaces.

An illustration of the theory is given to the construction of represenations of suitable infinite dimensional Heisenberg groups.

The thesis is very well presented and written in four Chapters. Chapter I contains prerequisites. Chapter II is devoted to the Aronszajn’s theory of reproducing kernels and functional Hilbert spaces. Chapter III is an exposition of inductive and projective limits of Hilbert spaces. Chapter IV is devoted to spaces of analytic functions on sequence spaces.

An illustration of the theory is given to the construction of represenations of suitable infinite dimensional Heisenberg groups.

Reviewer: H.Hogbe-Nlend

##### MSC:

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

46M40 | Inductive and projective limits in functional analysis |

46E20 | Hilbert spaces of continuous, differentiable or analytic functions |