Extension of Pluriharmonic Functions in Locally Convex Spaces

This chapter concerns the pluriharmonic extension on a domain over a complex manifold, the dimension of which may be infinite and for which the Levi problem may not have an affirmative solution. It discusses the envelopes of pluriharmony, and shows how to construct the envelope of pluriharmony of a family of pluriharmonie functions by the method of B. Malgrange. The chapter describes pseudoconvex hull. In finite dimensional case, J. Kajiwara had defined a pseudoconvex hull of a domain over a holomorphically convex manifold. The chapter presents the proposition in the infinite dimensional case, as given by Y. Matsuda.