Weighted inductive and projective limits of normed Köthe function spaces

Abstract

We consider locally convex inductive and projective limits of weighted normed Köthe function spaces of.measurable functions on a σ-finite measure space (X, Σ,μ). This general framework includes inductive and projective limits of weighted Lp-spaces, 1 ≤ p ≤ ∞, Lorentz spaces L p,q 1 < p < ∞, 1 ≤ q ≤ ∞, and Orlicz function spaces. In the projective case we characterize the quasinormability of L p(A) in terms of the increasing sequence A = (a n)n∈ℕ of weights on X. For the most important normed Köthe function spaces Lp, like Lp-spaces or L p,q-spaces, this condition on A = (an)n∈ℕ (V = (v n = 1/an)n∈ℕ) allows a useful projective description of the inductive limit topology of k p(V) by an associated system ofweights. A characterization of the Montel spaces L P(A) in terms of A = (an)n∈ℕ is also given. As an application we obtain functional analytic results on weighted inductive limits of spaces of holomorphic and harmonic functions.