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In this paper a direct proof of
the following theorem of Orlicz, Pettis, and Grothendieck is given.
Theorem 1. In a locally convex Hausdorff space each subseries of a series
converges with respect to the initial topology of the space if and only if each
subseries of the series converges with respect to the weak topology of the
space.
In a second theorem each of three additional conditions is shown to be equivalent
to subseries convergence in complete locally convex Hausdorff spaces. Two of these
equivalence are known for Banach spaces. The third condition, a weak compactness
condition on the unordered partial sums of the series, is new even for Banach spaces.
It is a consequence of the first theorem that a weak unconditional basis for a weak
sequentially complete locally convex Hausdorff space is an unconditional
basis.
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