Hölder's Inequality

Theorem

Hölder's Inequality for Integrals

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p, q \in \R_{>0}$ such that $\dfrac 1 p + \dfrac 1 q = 1$.

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Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space.

Then their pointwise product $f g$ is $\mu$-integrable, that is:

$f g \in \map {\LL^1} \mu$

and:

$\ds \norm {f g}_1$

$=$

$\ds \int \size {f g} \rd \mu$

$\ds $

$\le$

$\ds \paren {\int \size f^p \rd \mu}^{1 / p} \paren {\int \size g^q \rd \mu}^{1 / q}$

$\ds $

$=$

$\ds \norm f_p \cdot \norm g_q$

where:

$\size {f g}$ denotes the absolute value function applied to the pointwise product of $f$ and $g$

the $\norm {\, \cdot \,}_p$ signify $p$-seminorms.

Hölder's Inequality for Sums

Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:

$\dfrac 1 p + \dfrac 1 q = 1$

Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.

Formulation $1$

Let $\mathbf x$ and $\mathbf y$ denote the vectors consisting of the sequences:

$\mathbf x = \sequence {x_n} \in {\ell^p}_\GF$

$\mathbf y = \sequence {y_n} \in {\ell^q}_\GF$

where ${\ell^p}_\GF$ denotes the $p$-sequence space in $\GF$.

Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.

Then:

$\mathbf x \mathbf y \in {\ell^1}_\GF$

and:

$\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$

where:

$\mathbf x \mathbf y := \sequence {x_n y_n}_{n \mathop \in \N}$

$\norm {\mathbf x \mathbf y}_1$ is the $1$-norm, also known as the taxicab norm.

Formulation $2$

Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be sequences in $\GF$ such that $\ds \sum_{k \mathop \in \N} \size {x_k}^p$ and $\ds \sum_{k \mathop \in \N} \size {y_k}^q$ are convergent.

Then:

$\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$

Source of Name

This entry was named for Otto Ludwig Hölder.