On the norm continuity of -valued Gaussian processes

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Abstract
Let be the Schwartz space of all rapidly decreasing functions on Rn, be the topological dual space of and for each positive integer p, be the space of all elements of which are continuous in the p-th norm defining the nuclear Fréchet topology of . The main purpose of the present paper is to show that if {Xt, t ∈ [0, + ∞]} is an -valued Gaussian process and for any fixed φ ∈ the real Gaussian process {Xt(φ), t ∈ [0, + ∞)} has a continuous version, then for any fixed T > 0 there is a positive integer p such that {Xt, t ∈ [0, T]} has a version which is continuous in the norm topology of .