Compactness in Function Spaces

Real Analysis · Axiom Academy

LESSON Compactness in Function Spaces In C([a,b]) , closed-and-bounded is not enough — the Arzelà–Ascoli theorem says you also need equicontinuity. 1. Closed and Bounded Is Not Enough In , any closed bounded set is compact: every sequence has a convergent subsequence. In C([a,b]) this fails. The closed unit ball is closed and bounded, yet it is not compact — we can build a sequence inside it with no uniformly convergent subsequence. Every bump has sup norm exactly 1 — all inside the unit ball …yet any two are a full distance 1 apart — no subsequence can converge 2. Equicontinuity: One δ for the Whole Family Each continuous function has, for a given , its own . A family is equicontinuous when a single works for every function at once. Move two inputs within and the output of every f in the family stays within — uniform control on the whole collection, not function by function. 3. Arzelà–Ascoli, and Why Fails uniformly bounded — one bound M holds for all f , and equicontinuous — one works for all f at once. The family on is the textbook failure. It passes the first test and fails the second: for every n and every x — a single bound M = 1 . The slope is , peaking at n . As the wiggle outruns any fixed . climbs from 0 to 1 over an interval of length , which shrinks to 0 . So for no single can hold the whole family — it is not equicontinuous. By Arzelà–Ascoli it is not relatively compact : no subsequence of converges uniformly. 4. Add Equicontinuity Back: A Family That Works

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