Let $\{x_i\}_{i\in I}$ as orthonormal set in Hilbert space,
Now define $x = \sum_{i\in I}a_ix_i$ , can we prove $a_i = (x,x_i)$? If $I$ is uncountable set?
I know if we further impose $\sum_{i\in I}|a_i|^2 <\infty$ then set of $i$ such that $a_i \ne 0$ is almost countable（then taking limit,due to continuous of inner product）,what about the case without this assumption?
Best Answer
It depends on your definition of "$x = \sum_{i \in I} a_i \, x_i$". I would say that for any reasonable definition, this series converges if and only if $\sum_{i \in I} |a_i|^2 < \infty$.