The choice of the function space E ( S ) instead of C ( S ) may seem uncommon at first, but in fact it allows the smooth incorporation of the finitedimensional theory on max-stable distributions into the functional setup. By a suitable choice of f ∈ E ( S ) we obtain for s$_{1}$, . . . , s$_{d}$ ∈ S and x = ( x$_{1}$, . . . , x$_{d}$ ) ≤ 0 For details on the functional domain of attraction, see Aulbach et al. (2013) and Aulbach et al. (2014). Note that in the multivariate context, where random vectors (rv) instead of stochastic processes are considered, (2) and (3) are equivalent. Recall that a max-stable df on R d is always continuous. ON FUNCTIONAL RECORDS AND CHAMPIONS 3 P ( η ≤ f ) = P ( η$_{s}$$_{1}$ ≤ x$_{1}$, . . . , η$_{s}$$_{d}$ ≤ x$_{d}$ ) = exp ( − E ( max i =1 $_{,...,d}$| x$_{i}$ | Z$_{s}$$_{i}$ )) , (2) max i =1 ,...,n X ( i ) − d$_{n}$ c$_{n}$ →$_{D}$ ϑ , (3) P ( max i =1 ,...,n X ( i ) − d$_{n}$ c$_{n}$ ≤ f ) = P ( X − d$_{n}$ c$_{n}$ ≤ f ) n →$_{n}$$_{→∞}$ P ( ϑ ≤ f ) . where the right-hand side is the de Haan-Resnick-Pickands representation of a mul- tivariate standard max-stable distribution function (df), cf. de Haan and Resnick (1977), Pickands (1981). The mapping ‖ x ‖$_{D}$ := E (max$_{i}$$_{=1}$$_{,...,d}$ | x$_{i}$ | Z$_{s}$$_{i}$ ), x ∈ R $^{d}$, defines a multivariate D -norm, see Falk et al. (2011) for details. A stochastic process ϑ with non-degenerate univariate margins which realizes in C ( S ) fulfills condition (1) iff there is a continuous process X in the max-domain of attraction of ϑ , i. e. there are some norming functions c$_{n}$ ∈ C $^{+}$( S ), d$_{n}$ ∈ C ( S ), n ∈ N , such that where X $^{(1)}$, X $^{(2)}$, . . . are iid copies of X and →$_{D}$ denotes convergence in distribu- tion, that is, weak convergence of the distributions in ( C ( S ) , ‖·‖$_{∞}$ ). For details, see e. g. de Haan and Ferreira (2006, Section 9.2). We shortly write X ∈ D ( ϑ ) for (2). Relation (2) implies in particular that X is in the functional domain of attraction of the max-stable process ϑ , that is, for all f ∈ E ( S ),