structural:reformatting
Infinite dimensions

The generalization of the above norms to an infinite number of components leads to ℓ p {\displaystyle \ell ^{p}} and L p {\displaystyle L^{p}} spaces for p ≥ 1, {\displaystyle p\geq 1\,,} with norms

‖ x ‖ p = (∑ i ∈ N | x i | p) 1 / p and ‖ f ‖ p, X = (∫ X | f (x) | p d x) 1 / p {\displaystyle \|x\|_{p}={\bigg (}\sum _{i\in \mathbb {N} }\left|x_{i}\right|^{p}{\bigg)}^{1/p}{\text{ and }}\ \|f\|_{p,X}={\bigg (}\int _{X}|f(x)|^{p}~\mathrm {d} x{\bigg)}^{1/p}}

for complex-valued sequences and functions on X ⊆ R n {\displaystyle X\subseteq \mathbb {R} ^{n}} respectively, which can be further generalized (see Haar measure). These norms are also valid in the limit as p → + ∞ {\displaystyle p\rightarrow +\infty }, giving a supremum norm, and are called ℓ ∞ {\displaystyle \ell ^{\infty }} and L ∞. {\displaystyle L^{\infty }\,.}

Any inner product induces in a natural way the norm ‖ x ‖:= ⟨ x, x ⟩. {\textstyle \|x\|:={\sqrt {\langle x,x\rangle }}.}

Other examples of infinite-dimensional normed vector spaces can be found in the Banach space article.

Generally, these norms do not give the same topologies. For example, an infinite-dimensional ℓ p {\displaystyle \ell ^{p}} space gives a strictly finer topology than an infinite-dimensional ℓ q {\displaystyle \ell ^{q}} space when p < q. {\displaystyle p<q\,.}