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The authors present definitions of feedback stabilizability and detectability that are weaker than those previously appearing in the literature. Stabilizability and detectability (in this sense) are sufficient to ensure an equivalence between internal and external stability for well-posed systems. Furthermore, stabilizability and detectability are shown to be preserved under feedback (and other) connections. These results are linked to present a justification of input-output controller design methods for systems with unbounded control and observation.