MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

The space of distributions of compact support on a manifold and that of distributions on a compact manifold (perhaps with boundary) are not metrisable as has been pointed out here. They are, however, what is called Silva spaces (inductive limits of sequences of Banach spaces with compact interlocking mappings) and these have the desired properties---continuity of linear mappings (even non-linear ones, I think) are determined on sequences----and many more. This was know to the portuguese mathematician J. Sebastiao e Silve (cedilla circumflex missing---can't manage it) who created the theory in the 50's. An accessible account can be found in the first volume of Koethe's (problem with an umlaut now) treatise on toplogical vector spaces. The case of the distributions on a non-compact manifold (e.g. the real line) is more intricate. This space is what I called a $DLF$-space (for obvious reasons) in my thesis (unpublished). In this case, however, help is at hand in the form of the concept of partitions of unity for inductive and projective limits of locally convex spaces (de Wilde). This implies that that we can get it as a complemented subspace of a countable product of Slva spaces. I suspect that this suffices to justify the sequential arguments but, to my knowledge, nobody has investiagted this in detail.
The space of distributions of compact support on a manifold and that of distributions on a compact manifold (perhaps with boundary) are not metrisable as has been pointed out here. They are, however, what is called Silva spaces (inductive limits of sequences of Banach spaces with compact interlocking mappings) and these have the desired properties---continuity of linear mappings (even non-linear ones, I think) are determined on sequences----and many more. This was know to the portuguese mathematician J. Sebastiao e Silve (cedilla missing---can't manage it) who created the theory in the 50's. An accessible account can be found in the first volume of Koethe's (problem with an umlaut now) treatise on toplogical vector spaces. The case of the distributions on a non-compact manifold (e.g. the real line) is more intricate. This space is what I called a $DLF$-space (for obvious reasons) in my thesis (unpublished). In this case, however, help is at hand in the form of the concept of partitions of unity for inductive and projective limits of locally convex spaces (de Wilde). This implies that that we can get it as a complemented subspace of a countable product of Slva spaces. I suspect that this suffices to justify the sequential arguments but, to my knowledge, nobody has investiagted this in detail.