Given a sequnce $y_i\in H$ ($H$ is Hilbert space over $\mathbb{C}$) which satisfies $$ ||\sum\beta_i y_i||<A $$ where $A$ is real constant and the latter inequality is true for any sequence of scalars $\{\beta_i\}$, $0\leq|\beta_i|\leq1$ which are zero except (maybe) finite number of indices.
I need to show that $\sum y_i$ converges in norm.
I tried to show that $\sum ||y_i|| < \infty$ (and than it is enough) however I can only show that $\sum ||y_i||^2 < \infty$:
The latter follows from the fact that for any finite number of elements $x_i\in H$ there are scalars $c_i$, |c_i|=1 such that $||\sum c_i x_i||^2\geq \sum ||x_i||^2$
how can I proceed from here?
thank you
Suppose $\sum y_i$ is not Cauchy. That means there exists an $\epsilon$ such that for any $N \in \mathbb N$, there exists $m,n > N$ such that $\|\sum_{i=m}^n y_i\| \ge \epsilon$. So there exists a sequence of numbers $m_1 \le n_1 < m_2 \le n_2 < m_3 \le n_3 < \dots$ such that $\|\sum_{i=m_k}^{n_k} y_i\| \ge \epsilon$.
However the argument you gave can be modified to show that $$ \sum_{k=1}^\infty \left\|\sum_{i=m_k}^{n_k} y_i\right\|^2 < \infty ,$$ and this is a contradiction.