✍ Scribed by Giga, Mi-Ho; Giga, Yoshikazu; Saal, Jürgen
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In this section we prove the Ascoli-Arzelà-type compactness theorem introduced in §1.3.2. The theorem is fundamental since a variety of compactness results on various function spaces follows. Here we give a detailed proof, since the case that the domain of definition of functions is not compact is usually not contained in elementary course books.The proof is elementary and standard and based on fundamental arguments, such as the diagonal argument. In §1.3.2 the domain of definition of functions is supposed to form a metric space. This assumption can be relaxed to a topological space.
Compact Domains of Definition
First we treat the case that the domain of definition is compact. There are several different ways to prove the Ascoli-Arzelà theorem. Here we prefer a direct proof that does not require a new concept.
Ascoli-Arzelà Theorem
Theorem. Let M be a compact set (more precisely, compact topological space) and K a subset of C(M ), where C(M ) denotes the space of continuous functions on M . (Note that in the present case we may identify C(M ) with C ∞ (M ).) The set K is relatively compact in C(M ) if and only if K is bounded and equicontinuous in C(M ).Proof. We concentrate on the essential part, i.e., that boundedness and equicontinuity imply relative compactness. The converse direction is left to the reader (Exercise 5.1). If M is empty, then K is also empty, so we may assume that M is nonempty.
📜 SIMILAR VOLUMES
We prove compactness properties of various sets of differential forms with bounds on their exterior derivatives. This gives simple proofs of the Federer-Fleming result on normal currents and of "compensated compactness" lemmas.