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Let [tex] H = \{(x_n)_n \subseteq \mathbb{R} | \sum_{n=1}^{\infty} x_n < \infty \}[/tex] and for $(x_n)_n \in H$ define

$$\|(x_n)_n\|_H = \sup_{n} \left|\sum_{k=0}^{n} x_k \right|$$

Prove that $H$ is complete. Is $H$ a Hilbert space?

What is the best way to prove $H$ is complete?

To prove it's a Hilbert space, is it enough to prove that $\|.\|_H$ satisfies the parallellogram law?

Thanks in advance!