Existence of Unconditional Bases in Spaces of Polynomials and Holomorphic Functions

Our main result shows that every Montel Köthe echelon or coechelon space E of order 1 < p ≤ ∞ is nuclear if and only if for every (some) m ≥ 2 the space (( m E ), τ 0 ) of m ‐homegeneus polynomials on E endowed with the compact‐open topology τ 0 has an unconditional basis if and only if the space (ℋ( E ), τ δ ) of holomorphic functions on E endowed with the bornological topology τ δ associated to τ 0 has an unconditional basis (for coechelon spaces τ δ equals τ 0 ). The main idea is to extend the concept of the Gordon‐Lewis property from Banach to Fréchet and (DF) spaces. In this way we obtain techniques which are used to characterize the existence of unconditional basis in spaces of m ‐th (symmetric) tensor products and, as a consequence, in spaces of polynomials and holomorphic functions.