- #1

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Suppose,

[tex]

\sum_{n=0}^\infty a_n \, g_n(x) = 0

[/tex]

Does it necessarily follow that [itex]a_n = 0 \: \forall n[/itex]? If so, please provide a proof. If not, a counterexample would be helpful. If not, can I deduce anything about the the coefficients?

A similar formula,

[tex]

f(x) g(y) = 0

[/tex]

only implies that each function must be a constant.

[tex]

\sum_n f_n(x) \, g_n(y) = 0

[/tex]

Under a sum, my guess is that we can't say anything about each of the functions.