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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., when it is 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" by Grothendieck (available also in english translation) and the first volume of " Topological vector spaces " by Köthe.

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., when it is 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" by Grothendieck (available also in english translation) and the first volume of " Topological vector spaces " by Köthe.

It is not clear to me what the second question of your title means but as regards the one in the text, the space of smooth functions with compact support has, of course, as a complete lcs, a canonical representation as a projective limit of Banach spaces but the seminorms involved are rather complicated. However, the following might be helpful: it is an inductive limit with partition of unity (de Wilde) and this can be used to obtain stronger results than are generally available. For instance, it is not just a quotient of the lcs direct sum of the component $F$ spaces but is even a COMPLEMENTED subspace (Keim, "Induktive und projektive Limiten mit Zerlegung der Einheit", Man. Math. 10(1973) 191-195).

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.

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., when it is 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" by Grothendieck (available also in english translation) and the first volume of " Topological vector spaces " by Köthe.

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.

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.