Probabilistic characterization of certain Banach spaces (Q1077673)

The author studies the relation between mean square convergence for random elements with values in a Banach space and the structure of the Banach space. One of the results is as follows. For a real separable Banach space X, the following assertions are equivalent: (a) X is isomorphic to a Hilbert space. (b) Let (\(\Omega\),\({\mathcal A},P)\) be a probability measure space and \((\xi_ n)_{n\geq 1}\) be a sequence of mean-zero random elements in \(L^ 2(\Omega;X)\). Then \(\xi_ n\) converges in mean square if (and only if) the following two conditions are satisfied; (i) for each \(f\in X^*(=the\) dual of X), \(<\xi_ n,f>\) converges in mean square, (ii) the set \(\{R_{\xi_ n}\}\) of covariance operators of \(\xi_ n\) is relatively compact with respect to the nuclear norm. The author also shows that assertion (b) holds only for Gaussian random elements if and only if X is of type 2, provided X has the approximation property.