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Let (E, i) be a real or complex locally convex (Hausdorff) space. We say that a subspace M of E is invariant for a collection. B of linear operators on E to itself if TM c M for each Tin 2'. A subspace M of E will be called proper if it is distinct from {o} and E. We define some special collections of linear operators in § 2 and consider the problem of existence of proper invariant subspaces for them in § 3 and § 4. I should like to express my gratitude to my research supervisor, Dr. F. Smithies, for his valuable criticism, kind advice and encouragement. I take this opportunity to thank the Association of Commonwealth Universities for a Commonwealth scholarship.