In a recent paper [4] we considered a semigroup of linear contractions or a group of linear isometries in a normed vector space, and obtained sufficient conditions for the orbits to he on parallel closed hyperplanes. In the present note, we take a more general viewpoint and shall prove some theorems on extension of continuous linear functionals which are invariant under a group or a semigroup of continuous linear maps on a locally convex topological vector space. These more general results

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... de those in [4] as direct consequences. Proof. If <p is the zero functional on L, then we may take the zero functional on E as \¡/. We shall therefore assume that <p =£ 0. Let M = (x G L: <p(x) = a). We claim that M n Int K = 0. Suppose that x0 G M n Int K. Let K be a circled neighborhood of 0 in E such that x0 + V E K.