Harmonic Analysis

We use τK to denote the topology of DK (Ω) equipped with such metric. The topology of D (Ω) can be defined precisely. Let β be the collection of all convex balanced sets W ⊂ D (Ω) such that DK (Ω) ∩ W ∈ τK for every compact K ⊂ Ω. Let τ be the collection of all unions of sets of the form φ+W with φ ∈ D (Ω) and W ∈ β. Theorem 1. τ is a topology in D (Ω) and β is a local base for τ . The topology τ makes D (Ω) into a locally convex topological vector space. Many important properties of D (Ω) are included in the following theorem. Theorem 2. (a) A convex balanced subset V of D (Ω) is open iff V ∈ β. (b) The topology τK of any DK (Ω) coincides with the subspace topology that DK inherits from D (Ω). (c) If E is bounded subset of D (Ω), then E ⊂ DK (Ω) for some compact K ⊂ Ω and for each N ≥ 0, there exists MN < ∞ s.t., ‖φ‖N ≤ MN for any φ ∈ E. (d) D (Ω) has the Heine-Borel property. (e) If {φi} is a Cauchy sequence in D (Ω), then {φi} ⊂ DK (Ω) for some compact K ⊂ Ω and lim i,j→∞ ‖φi − φj‖N = 0 for any N ≥ 0. (f) If φi → 0, then {φi} ⊂ DK (Ω) for some compact K ⊂ Ω and Dφi → 0 uniformly for every multi-index α. (g) In D (Ω), every Cauchy sequence converges. Theorem 3. Every differential operator D : D (Ω) → D (Ω) is continuous. Definition 1. A continuous linear functional on D (Ω) is called a distribution. The space of all distributions in Ω is denoted by D′ (Ω).