\(L^p\) spaces (sometimes known as Lebesgue Spaces) are function spaces, i.e. collections of functions that all have the same domain and codomain that have properties in common, defined using a natural generalization of the Lp Norm. They form an important class of [[banarch-space|Banarch Spaces]] in functional analysis and of [[topological-vector-spaces]].
For any given positive real number \(p>0\), let us denote be \(L^p\) the family of functions \(f:\Omega \to \mathbb{R}\) on the measurable space \((\Omega, \mathcal{F}, \mu)\) with \(|f|^p \in L^1\), or equivalently \[\|f\|_{p}:=\left( \int _{\Omega}|f|^p \, d\mu \right)^{1/p}<\infty.\]
Theorem: \(L^p\) space is a real vector space closed under addition and scalar multiplication. Proof: Since \(|f+g|^p\leq (2\cdot \max(|f|, |g|))^p\leq 2^p(|f|^p+|g|^p)\) we have that \[ \|f+g\|_{p}^p\leq 2^p(\|f\|_{p}^p+\|g\|_{p}^p) \] for every \(f,g \in L^p\). This implies that if \(f,g \in L^p\), then \(f+g \in L^p\). Also, for every \(c \in \mathbb{R}\) and \(f \in L^p\) we have \(\|cf\|_{p}=|c|\cdot \|f\|_{p}\) and so if \(f \in L^p\) then \(cf \in L^p\) for every constant \(c \in \mathbb{R}\).
Theorem (Completeness) The space \(L^p\) is complete for any \(p \in[1, \infty]\), that is, for any Cauchy sequence \(\{f_{n}, n \geq 1\}\subseteq L^p\) (i.e. with the property that for every \(\epsilon>0\) there is an integer \(n_\epsilon\) such that \(\|f_{n}-f_{m}\|_{p}\leq \epsilon\) whenever \(n\geq n_{\epsilon}\)) there is a unique \(f \in L^p\) such that \[ \|f_{n}-f\|_{p}\stackrel{n\to \infty}{\to}0. \] Moreover, there exists a subsequence \(\{f_{n_{k}},k\geq 1\}\subseteq \{f_{n},n\geq 1\}\), as well as a function \(F:\Omega \to[0,\infty)\) in \(L^p\) such that for \(\mu\)-a.e., \(\omega \in \Omega\) we have \[ |f_{n_{k}}(\omega)|\leq F(\omega);\quad \&\quad \lim_{ k \to \infty } f_{n_{k}}(\omega)=f(\omega). \] Proof: