Completeness of C([a,b])

Real Analysis · Axiom Academy

LESSON Completeness of C([a,b]) The continuous functions on a closed interval, measured by the sup norm, form a complete space — every Cauchy sequence converges to a function that is still continuous. 1. The Space C([a,b]) and the Sup Norm C([a,b]) is the set of all continuous real-valued functions on the closed interval [a,b] . To talk about functions getting "close," we need a way to measure the size of a function. The supremum norm takes the single largest height the function reaches: The norm is the tallest deviation from zero across the whole interval Because a continuous function on a closed bounded interval is bounded and actually attains its extremes (Extreme Value Theorem), this supremum is always finite — so every has a well-defined norm. 2. Cauchy Means Uniformly Close A sequence is Cauchy in this norm if its terms eventually get uniformly close to each other. The sup norm makes "close" mean close everywhere at once : the worst gap between any two late terms is small. Geometrically, beyond index N every graph is trapped inside a tube of half-width around the others. As shrinks, the curves are pressed together with nowhere to spread — a uniformly Cauchy family. 3. Pointwise vs Uniform Convergence Completeness hinges on the difference between two kinds of convergence. The sup norm delivers the strong one. For each fixed x , . The chosen N may depend on x — some points settle fast, others slowly.

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