Definition:P-Sequence Metric/Real Sequences

Definition
Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $d_p: A \times A: \to \R$ be the real-valued function defined as:
 * $\displaystyle \forall x = \left\langle{x_i}\right\rangle, y = \left\langle{y_i}\right\rangle \in A: d_p \left({x, y}\right) := \left({\sum_{k \mathop \ge 0} \left\vert{x_k - y_k}\right\vert^p}\right)^{\frac 1 p}$

The metric space $\left({A, d_p}\right)$ is the $p$-sequence space on $\R$ and is denoted $\ell^p$.

Also see

 * Definition:Sequence Space


 * $p$-Sequence Space of Real Sequences is Metric Space