The definition of a total set according to the Encyclopedia of Mathematics is:
A set $M⊂T$, where $T$ is a topological vector space, is called a total set or fundamental set if the linear span of $M$ is dense in $T$.
If $M$ is already dense in $T$, does that ensure that it is also total in $T$?
Context: I am looking at a proof where a set is assumed to be total in another, however the author works only with the density of the supposedly total set, and not with the closure of it's linear span. This has lead me to wonder whether it is sufficient to work the density of the original set alone, and not with the closure of it's linear span.
The linear span of a set contains the set itself, so the linear span of a dense set has a dense subset, so it is also dense.