Differentiable paths in topological vector spaces

In this note, we show that a strong form of the Bolzano-Weierstrass theorem in  a topological vector space E[ T] is equivalent, for example, to the assertion that there are enough differentiable paths,  x (t),  with non-trivial tangen t vectors, so that a function f defined  on E will be sequentially continuous for T if the composites f(x(t)) are all continuous. For a large class of locally convex spaces, this property is shown to be equivalent to the statement that the bounded sets of E[T] are finite dimensional. This leads to some very precise results for special cases.