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This paper presents a fairly general version of the well-knownGleason-Kahane-$\dot{\text{Z}}$elazko (GKZ) theorem in the spirit of a GKZ typetheorem obtained recently by Mashreghi and Ransford for Hardy spaces. Ineffect, we characterize a class of linear functionals as point evaluations onthe vector space of all complex polynomials $\cl P$. We do not make anytopological assumptions on $\cl P$. We then apply this characterization topresent a version of the GKZ theorem for a vast class of topological spaces ofcomplex-valued functions including the Hardy, Bergman, Dirichlet, and many morewell-known function spaces. We obtain this result under the assumption ofcontinuity of the linear functional, which we show, with the help of anexample, to be a necessary condition for the desired conclusion. Lastly, we usethe GKZ theorem for polynomials to obtain a version of the GKZ theorem forstrictly cyclic weighted Hardy spaces.

Multiplicativity of linear functionals on function spaces on an open unit disc