******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces?
I have this assignment question from Functional Analysis class stating:
Let $ \mathcal{H} $ be a Hilbert Space with a sequence $ \{x_n\}_{n=1}^{\infty} $ of elements in $ \mathcal{H} $. There is a constant $ A \in R $ such that for every sequence $ \{a_n\}_{n=1}^{\infty} $ satisfying $ 0 \leq |a_n| \leq 1 $ all zero except a finite number of elements, we have ||$ \sum_{n} a_nx_n $|| $ \leq $ A. We are asked to prove $\lim_{N\to\infty} {\sum_{n=1}^{N} x_n}$ exists in the norm. Also, does this hold if the space H is not complete? What about a norm space where the norm is not induced from an inner product?
Last part before this I proved:
For all $ \{ x_n \}_{n=1}^N \in \mathcal{H} $ there are scalars $ \{ a_n \}_{n=1}^N $ on the complex unit circle such that : $||\sum_{n=1}^{N} a_nx_n ||^2 \geq \sum_{n=1}^{N} ||x_n||^2 $
I don't exactly know how to incorporate that previous part or if I even need to for that matter. I would really appreciate help.
PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces?