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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?

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    $\begingroup$ 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 $k$. $\endgroup$ Commented Jan 28, 2022 at 17:46
  • $\begingroup$ @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 $\endgroup$
    – Gob
    Commented Jan 28, 2022 at 21:19
  • $\begingroup$ @Gob What is your definition of such a function series? $\endgroup$
    – Gary
    Commented Jan 29, 2022 at 6:04
  • $\begingroup$ @ I added an answer $\endgroup$
    – Gob
    Commented Jan 29, 2022 at 12:10

1 Answer 1

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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.

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