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We construct normed spaces of real-valued functions with controlled growth onpossibly infinite-dimensional state spaces such that semigroups of positive,bounded operators $(P_t)_{t\ge 0}$ thereon with $\lim_{t\to 0+}P_t f(x)=f(x)$are in fact strongly continuous. This result applies to prove optimal rates ofconvergence of splitting schemes for stochastic (partial) differentialequations with linearly growing characteristics and for sets of functions withcontrolled growth. Applications are general Da Prato-Zabczyk type equations andthe HJM equations from interest rate theory.

A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations