Let C(X) be the Banach space of continuous real-valued functions of an infinite compactum X with the sup-norm, which is homeomorphic to the pseudo-interior s = (-1, 1) ω of the Hilbert cube Q = [-1, 1] ω . We can regard C(X) as a subspace of the hyperspace exp(X x R) of nonempty compact subsets of X x R endowed with the Vietoris topology, where R = [-~, ~] is the extended real line (cf. (Fedorchuk, 1991)). Then the closure C(X) of C(X) in exp(X x R) is a compactification of C(X). We show that the pair (C(X), C(X)) is homeomorphic to (Q, s) if X is locally connected. As a corollary, we give the affirmative answer to a question of Fedorchuk (Fedorchuk, 1996, Question 2.6).