Hilbert Sequence Space is Metric Space/Proof 1
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Theorem
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a metric space.
Proof
$\ell^2$ is a particular instance of the general $p$-sequence space $\ell^p$.
Hence $p$-Sequence Space of Real Sequences is Metric Space can be applied directly.
$\blacksquare$