I know there have been a number of questions on Hilbert spaces and orthonormal basis, but I can't find any answers to these two questions:
1) Let $H$ be a Hilbert space, and say we found a Hilbert basis by taking a maximal orthonormal set. This question is only interesting if we assume the set of basis vectors is uncountable, so let's do that. I know how to show that the finite linear span of this set is dense in H. But does this imply that every vector in H is a (countably) infinite linear combination of our basis vectors? I know we can approach arbitrarily close using finite linear combinations, but the basis vectors being used in the finite linear combinations may keep changing.
2) My textbook writes expressions such as $\sum_{k=1}^{\infty}\langle v_k,h\rangle v_k$ where $v_k$ forms an orthonormal set in $H$. My question here is: How can we say that the order of the summation does not matter? I know that Bessel's Inequality says $\sum_{k=1}^{\infty}\langle v_k,h\rangle^2$ converges, but I not sure if this helps, or how it helps.
Best Answer
1) The sum of over an uncountable set of indices is defined precisely as the limit over the net of finite sets, ordered by inclusion.
2) Order does not matter because the convergence is absolute. You have$$ \left\|\sum_{k>N}^M\langle v_k,h\rangle\,v_k\right\|^2=\sum_{k=N+1}^M|\langle v_k,h\rangle| $$ and so Bessel's inequality, as you mention, implies that order does not matter, with the exact same proof as the one that absolute convergence implies that order does not matter for sequences of numbers.
Your two questions are related, because defining series as in 1), in the countable case, is precisely equivalent to absolute convergence.