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The specific space that you mention---smooth functions with compact support---is not a $DF$ space. Under certain situations, an $LF$ space can be a $DFN$ space, e.g., a strict inductive limit of finite dimensional spaces. In general, a (strict) $LF$ space as introduced in the seminal article by Dieudonné and Schwartz is not a $DF$ space but it can be in special circumstances, e.g., if it is the strict inductive limit of a sequence of Banach spaces. Good references are "Espaces vectorielles topologiques" (available also in english translation) and the first volume of " Topological vector spaces " by Köthe.