Sequence space
ℓ spaces are increasing in p

For p ∈ {\displaystyle p\in }, the spaces ℓ p {\displaystyle \ell ^{p}} are increasing in p {\displaystyle p}, with the inclusion operator being continuous: for 1 ≤ p < q ≤ ∞ {\displaystyle 1\leq p<q\leq \infty }, one has ‖ x ‖ q ≤ ‖ x ‖ p {\displaystyle \|x\|_{q}\leq \|x\|_{p}}. Indeed, the inequality is homogeneous in the x i {\displaystyle x_{i}}, so it is sufficient to prove it under the assumption that ‖ x ‖ p = 1 {\displaystyle \|x\|_{p}=1}. In this case, we need only show that ∑ | x i | q ≤ 1 {\displaystyle \textstyle \sum |x_{i}|^{q}\leq 1} for q > p {\displaystyle q>p}. But if ‖ x ‖ p = 1 {\displaystyle \|x\|_{p}=1}, then | x i | ≤ 1 {\displaystyle |x_{i}|\leq 1} for all i {\displaystyle i}, and then ∑ | x i | q ≤ ∑ | x i | p = 1 {\displaystyle \textstyle \sum |x_{i}|^{q}\leq \textstyle \sum |x_{i}|^{p}=1}.
detectable_format:number_bullet_lists[4]