فضای Lp

x = (x1, x2, ..., xn)

$\left\|x\right\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.$

$\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.$

$\left\|x\right\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}$

$\left\|x\right\|_{2}\leq \left\|x\right\|_{1}.$

$\left\|x\right\|_{1}\leq {\sqrt {n}}\left\|x\right\|_{2}.$

$\left\|x\right\|_{p}\leq \left\|x\right\|_{r}\leq n^{(1/r-1/p)}\left\|x\right\|_{p}.$

$d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}$

(Rn, dp) = ℓnp

$(x_{n})\mapsto \sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}},$

${\begin{aligned}&(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots )+(y_{1},y_{2},\ldots ,y_{n},y_{n+1},\ldots )\\={}&(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\ldots ),\\[6pt]&\lambda \cdot \left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\\={}&(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n},\lambda x_{n+1},\ldots ).\end{aligned}}$

$\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\cdots \right)^{1/p}$

$\left\|x\right\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|,|x_{n+1}|,\ldots )$

$\left\|x\right\|_{\infty }=\lim _{p\to \infty }\left\|x\right\|_{p}$

$\ell ^{p}(I)=\left\{(x_{i})_{i\in I}\in \mathbb {K} ^{I};\,\sum _{i\in I}|x_{i}|^{p}<\infty \right\}\,$

$\left\|x\right\|_{p}=\left(\sum _{i\in I}|x_{i}|^{p}\right)^{1/p}$

$\|f\|_{p}\equiv \left(\int _{S}|f|^{p}\;\mathrm {d} \mu \right)^{1/p}<\infty$

${\begin{aligned}(f+g)(x)&=f(x)+g(x),\\(\lambda f)(x)&=\lambda f(x)\end{aligned}}$

$\|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right).$

$\|f\|_{\infty }\equiv \inf\{C\geq 0:|f(x)|\leq C{\text{ for almost every }}x\}.$

$\|f\|_{\infty }=\lim _{p\to \infty }\|f\|_{p}.$

${\mathcal {N}}\equiv \{f:f=0\ \mu {\text{-almost everywhere}}\}=\ker(\|\cdot \|_{p})\qquad \forall \ 1\leq p<\infty$

$L^{p}(S,\mu )\equiv {\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}$

$\langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x)$ $f\mapsto \kappa _{p}(g)(f)=\int fg\,\mathrm {d} \mu \ \$

$j_{p}:L^{p}(\mu ){\overset {\kappa _{q}}{\longrightarrow }}L^{q}(\mu )^{*}{\overset {\left(\kappa _{p}^{-1}\right)^{*}}{\longrightarrow }}L^{p}(\mu )^{**}$

$\ \|\mathbf {1} f^{p}\|_{1}\leq \|\mathbf {1} \|_{q/(q-p)}\|f^{p}\|_{q/p}$

leading to

$\ \|f\|_{p}\leq \mu (S)^{1/p-1/q}\|f\|_{q}$ .

$\|I\|_{q,p}=\mu (S)^{1/p-1/q}$

1 ≤ p < ∞.

(S, Σ, μ)

$f=\sum _{j=1}^{n}a_{j}\mathbf {1} _{A_{j}}$

Aj ∈ Σ ${\mathbf {1} }_{A_{j}}$ $A_{j}$

$F\subset A\subset U\subset V\quad {\text{and}}\quad \mu (U)-\mu (F)=\mu (U\setminus F)<\varepsilon$

0 ≤ φ ≤ 1 on S that is 1 on F and 0 on S ∖ U, with

$\int _{S}|\mathbf {1} _{A}-\varphi |\,\mathrm {d} \mu <\varepsilon \ .$

$\forall f\in L^{p}(\mathbf {R} ^{d}):\qquad \left\|\tau _{t}f-f\right\|_{p}\to 0,\quad {\text{ as }}\mathbf {R} ^{d}\ni t\to 0,$

$(\tau _{t}f)(x)=f(x-t).$

Lp (0 < p < 1)[edit]

$N_{p}(f)=\int _{S}|f|^{p}\,d\mu <\infty .$

As before, we may introduce the p-norm || f ||p = Np( f )1/p, but || · ||p does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality (a + b) p ≤ a p + b p, valid for a, b ≥ 0 implies that (Rudin 1991, §1.47)