# Series with negative indexes

Recall the well known theorem:

Theorem. Consider a series $$\sum_{k=0}^{\infty} g_k$$ of functions on a set $$S\subset\Bbb R.$$ Suppose that each $$g_k$$ is continuous on $$S$$ and that the series converges uniformly on $$S.$$ Then the series $$\sum_{k=0}^{\infty} g_k$$ represents a continuous function on $$S.$$

Is it correct even if we replace $$\sum_{k=0}^{\infty} g_k$$ by $$\sum_{k\in\Bbb Z} g_k$$ in the Theorem above?

Actually, I brought this theorem as just an example but I can make it more general. Once a fact holds for the series with positive Indexes does it hold for all integers?

• I think it holds, but you don't need negative indexes. That theorem holds for sets sets of functions $f_k$ as well, supposing they all satisfy the continuity criteria and such. So adding two series should yield the same result as adding one series with negative indexes. That could change if the $g_k$ are dependent on $k$. Commented Jan 28, 2022 at 17:46
• @TurlocTheRed, if I understood correctly this hold for countable set then the theorem for countable set of functions that satisfies the conditions. If this the case, how about uncountable set of continuous function
– Gob
Commented Jan 28, 2022 at 21:19
• @Gob What is your definition of such a function series?
– Gary
Commented Jan 29, 2022 at 6:04
If the $$\sum _{k} g_k$$ satisfies the hypothesis of the above theorem then the conclusion holds. The argument is the same as for original theorem.