A proof of Cartan's theorems $A$ and $B$

Abstract

The known proofs of these theorems depend on one of the following : (i) Cartan's Lemma of invertible holomorphic matrices ([2], [3]), (ii) methods of partial differential equations ([5]), and (iii) methods of differential geometry ([!]). In the proof here essentially we make use only of Dolbeault's Lemma (I.D.3, [4]) and Schwartz's Theorem (App.B, 12, [4]). Theorem B is first proved and then Theorem A is derived from it.