For eagles only: probably the most difficult proof of the Arzelà-Ascoli theorem – via the Stone-Čech compactification

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Let C(K) denote the Banach space of all (real or complex) continuous functions on a compact Hausdorff space K.We present a novel point of view on the classical Arzelà-Ascoli theorem: For every pointwise bounded and equicontinuous subset of C(K) there is a continuous mapping J : β → C(K), where β denotes the Stone-Čech compactification of , such that ⊂ J (β); hence the closure of is compact.