Levi Extension Theorems for Meromorphic Functions of Weak Type in Infinite Dimension

The aim of paper is to give some results, that prepare for studying the problem on cross theorems for separately ( · , W ) -meromorphic functions. Some general versions of extension theorem of Levi type are extended to the classes of meromorphic functions f on D × ( Δ r \ Δ ¯ ) with values in a locally convex space F . Here, the function f is assumed that, for each z ∈ D ∗ , the function f z = f ( z , · ) has a ( F , W )-meromorphic extension to Δ r , where F is either a locally (or sequentially) complete locally convex space or a Fréchet space, the space W ⊆ F ′ is separating (or determines boundedness), Δ r = { λ ∈ C : | λ | < r } with r > 1 , Δ = Δ 1 and D is either a domain in C n or a balanced domain in a Fréchet space containing a non-pluripolar balanced convex compact subset, D ∗ is dense in D .