Distance to spaces of continuous functions

We link here distances between iterated limits, oscillations, and distances to spaces of continuous functions. For a compact space K, a uniformly bounded set H   of the space of real-valued continuous functions C(K)C(K), and ε⩾0ε⩾0, we say that H ε-interchanges limits with K, if the inequality|limnlimmfm(xn)−limmlimnfm(xn)|⩽ε holds for any two sequences (xn)(xn) in K   and (fm)(fm) in H, provided the iterated limits exist. We prove that H ε-interchanges limits with K if and only if the inequality for the oscillationsosc∗(f)=supx∈Kosc∗(f,x)=supx∈Kinf{supy∈U|f(y)−f(x)|:Uneighb. of x}⩽ε, holds for every f   in the closure clRK(H)clRK(H) of H   in RKRK. Since oscillations actually measure distances to spaces of continuous functions, we get that if H ε-interchanges limits with K, thendˆ(clRK(H),C(K)):=supf∈clRK(H)d(f,C(K))⩽ε. Conversely, if dˆ(clRK(H),C(K))⩽ε, then H 2ε-interchanges limits with K. We also prove that H ε-interchanges limits with K   if and only if its convex hull conv(H)conv(H) does. As a consequence we obtain that for each uniformly bounded pointwise compact subset H   of RKRK we havedˆ(clRK(conv(H)),C(K))⩽5dˆ(H,C(K)). The above estimates can be applied to measure distances from elements of the bidual E**E** to the Banach space E  : for a w*w*-compact subset H   of E**E**, we have dˆ(w*-cl(conv(H)),E)⩽5dˆ(H,E). These results are quantitative   versions of the classical Eberlein–Grothendieck and Krein–Smulyan theorems. In the case of Banach spaces these quantitative generalizations have been recently studied by M. Fabian, A.S. Granero, P. Hajék, V. Montesionos, and V. Zizler. Our topological approach allows us to go further: most of the above statements remain true for spaces C(X,Z)C(X,Z) for a paracompact (in some cases, a normal countably compact) space X and a convex compact subset Z of a Banach space.