$ℓ^{p} (Z_{n})$ spaces

Given a natural number $N$ , the space $ℓ^{p} (Z_{N})$ is defined as:

ℓ^{p} (Z_{N}) = {z = (z_{0}, z_{1}, \dots, z_{N - 1}) \in C^{N} | \sum_{j = 0}^{N - 1} | z_{j} |^{p} < \infty}

This space is a vector subspace of $C^{N}$ , equipped with the norm:

∥ z ∥_{p} = {(\sum_{j = 0}^{N - 1} | z_{j} |^{p})}^{1 / p}

Remarks

Although $ℓ^{p} (Z_{N})$ is algebraically identical to $C^{N}$ , they are Banach spaces.
When $p = 2$ , it gains the structure of a Hilbert space, with an inner product defined by:

⟨ z, w ⟩ = \sum_{j = 0}^{N - 1} z_{j} \overset{―}{w_{j}}