This is a response to the above comment but is too long for a comment. ad 1. This is a corollary of the fact that $L^1$ is the dual of $L^\infty$ with the so-called strict topology, i.e., the finest locally convex topology which agrees with the $L^1$ topology on the unit ball. This essentially goes back to Saks (T.A.M.S. 35 (1933) 549-556) and a modern approach in the secondary literature can be found in the book "Saks spaces and applications to functional analysis". ad 2. yes. ad 3. no, but a variant is true. You use sequences of measures which converge weakly to zero in an analogous manner. This follows from the fact that the space of continuous functions, say on a compact interval, is the dual of the space of measures thereon with the so-called bounded weak star topoloy

This is a response to the above comment but is too long for a comment. ad 1. This is a corollary of the fact that $L^1$ is the dual of $L^\infty$ with the so-called strict topology, i.e., the finest locally convex topology which agrees with the $L^1$ topology on the unit ball. This essentially goes back to Saks (T.A.M.S. 35 (1933) 549-556) and a modern approach in the secondary literature can be found in the book "Saks spaces and applications to functional analysis". ad 2. yes. ad 3. no, but a variant is true. You use sequences of smooth functios which tend to zero for the weak topology induced by the continuous functions. This follows from the fact that the space of continuous functions, say on a compact interval, is the dual of the space of measures thereon with the so-called bounded weak star topology.