As topological vector spaces are uniform spaces, it is appropriate to discuss completeness. As with a uniform space, a topological vector space is complete if it has no holes: everything that should be there actually is there.
Where this gets interesting is in the question as to what should be there. To determine this, one has to have some method of discovering holes. This is usually done by means of Cauchy nets, but in a given application it may not be necessary that all holes be filled and this leads to weaker notions of completeness.
In strict order of decreasing strength, we have the following notions of completeness.
Sequentially complete implies locally complete because every locally Cauchy sequence is a Cauchy sequence. The inverse implication “locally complete” sequentially complete is true for example in metrizable locally convex topological vector spaces, but not in general: A Cauchy sequence will not be locally Cauchy in general.
The problem to precisly characterize the spaces in which every convergent sequence is locally convergent is an open problem according to Köthe volume 1 (which is quite old, so it could have been solved in the meantime).