  We provide a detailed analysis of the Gelfand integral on Fr\'echet spaces,
showing among other things a Vitali theorem, dominated convergence and a Fubini
result. Furthermore, the Gelfand integral commutes with linear operators. The
Skorohod integral is conveniently expressed in terms of a Gelfand integral on
Hida distribution space, which forms our prime motivation and example. We
extend several results of Skorohod integrals to a general class of pathwise
Gelfand integrals. For example, we provide generalizations of the
Hida-Malliavin derivative and extend the integration-by-parts formula in
Malliavin Calculus. A Fubini-result is also shown, based on the commutative
property of Gelfand integrals with linear operators. Finally, our studies give
the motivation for two existing definitions of stochastic Volterra integration
in Hida space.
