Functional Spaces

Abstract

Let Ω be a domain of the (n + 1)-dimensional space ℝ^{n +1} and φ(x, t) a function continuous in Ω. The closure in Ω of the set of all points (x, t) for which φ(x, t) ≠ 0 is called the support of the function φ(x,t) in Ω and denoted by supp φ(х, t). For an integer q ≥ 0 we denote by C ^q (Ω) [resp., C ^q $(\bar{\Omega })$] the set of all functions φ(х,t) continuous in Ω [resp., $(\bar{\Omega })$] together with their derivatives of order less than or equal to q. C ^q₀ (Ω) [resp., C ^∞₀ (Ω)] will designate the set of functions φ(x, t) ∈ C ^q (Ω) [resp., C ^∞(Ω)] with supports in Ω.