Derived operator plays an important role in describing algebraic properties of many mathematical structures such as topology, matroid, convergence and convex space. In this paper, we present the notions of L-concave derived internal relation space and L-convex derived enclosed relation space by which we characterize L-concave space and L-convex space. Based on this, we further introduce some other structures such as L-concave derived hull space and L-convex derived hull space. We find that these spaces are isomorphic to L-concave space and L-convex space.