Vector space
Banach spaces

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.

A first example is the vector space ℓ p {\displaystyle \ell ^{p}} consisting of infinite vectors with real entries x = (x 1, x 2, …, x n, …) {\displaystyle \mathbf {x} =\left(x_{1},x_{2},\ldots,x_{n},\ldots \right)} whose p {\displaystyle p} -norm (1 ≤ p ≤ ∞) {\displaystyle (1\leq p\leq \infty)} given by ‖ x ‖ ∞:= sup i | x i | for p = ∞, and {\displaystyle \|\mathbf {x} \|_{\infty }:=\sup _{i}|x_{i}|\qquad {\text{ for }}p=\infty,{\text{ and }}} ‖ x ‖ p:= (∑ i | x i | p) 1 p for p < ∞. {\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i}|x_{i}|^{p}\right)^{\frac {1}{p}}\qquad {\text{ for }}p<\infty.}

The topologies on the infinite-dimensional space ℓ p {\displaystyle \ell ^{p}} are inequivalent for different p. {\displaystyle p.} For example, the sequence of vectors x n = (2 − n, 2 − n, …, 2 − n, 0, 0, …), {\displaystyle \mathbf {x} _{n}=\left(2^{-n},2^{-n},\ldots,2^{-n},0,0,\ldots \right),} in which the first 2 n {\displaystyle 2^{n}} components are 2 − n {\displaystyle 2^{-n}} and the following ones are 0, {\displaystyle 0,} converges to the zero vector for p = ∞, {\displaystyle p=\infty,} but does not for p = 1: {\displaystyle p=1:} ‖ x n ‖ ∞ = sup (2 − n, 0) = 2 − n → 0, {\displaystyle \|\mathbf {x} _{n}\|_{\infty }=\sup(2^{-n},0)=2^{-n}\to 0,} but ‖ x n ‖ 1 = ∑ i = 1 2 n 2 − n = 2 n ⋅ 2 − n = 1. {\displaystyle \|\mathbf {x} _{n}\|_{1}=\sum _{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.}

More generally than sequences of real numbers, functions f: Ω → R {\displaystyle f:\Omega \to \mathbb {R} } are endowed with a norm that replaces the above sum by the Lebesgue integral ‖ f ‖ p:= (∫ Ω | f (x) | p d μ (x)) 1 p. {\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,{d\mu (x)}\right)^{\frac {1}{p}}.}

The space of integrable functions on a given domain Ω {\displaystyle \Omega } (for example an interval) satisfying ‖ f ‖ p < ∞, {\displaystyle \|f\|_{p}<\infty,} and equipped with this norm are called Lebesgue spaces, denoted L p (Ω). {\displaystyle L^{\;\!p}(\Omega).}

These spaces are complete. (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.) Concretely this means that for any sequence of Lebesgue-integrable functions f 1, f 2, …, f n, … {\displaystyle f_{1},f_{2},\ldots,f_{n},\ldots } with ‖ f n ‖ p < ∞, {\displaystyle \|f_{n}\|_{p}<\infty,} satisfying the condition lim k, n → ∞ ∫ Ω | f k (x) − f n (x) | p d μ (x) = 0 {\displaystyle \lim _{k,\ n\to \infty }\int _{\Omega }\left|f_{k}(x)-f_{n}(x)\right|^{p}\,{d\mu (x)}=0} there exists a function f (x) {\displaystyle f(x)} belonging to the vector space L p (Ω) {\displaystyle L^{\;\!p}(\Omega)} such that lim k → ∞ ∫ Ω | f (x) − f k (x) | p d μ (x) = 0. {\displaystyle \lim _{k\to \infty }\int _{\Omega }\left|f(x)-f_{k}(x)\right|^{p}\,{d\mu (x)}=0.}

Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.
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