Hölder's Inequality (Special Case)

Theorem

Let $p > 1$ be a real number, and let $q > 1$ be a real number such that $\dfrac 1 p + \dfrac 1 q = 1$.

Let $\mathbf{x} = \langle {x_n} \rangle$ and $\mathbf{y} = \langle {y_n} \rangle$ be members of the Lebesgue spaces $\ell^p$ and $\ell^q$, respectively.

Let $\left\Vert {\mathbf{x}} \right\Vert_p$ denote the $p$-norm of $\mathbf{x}$.

Then $\left\Vert {\mathbf{x}\mathbf{y}} \right\Vert_1 \le \left\Vert {\mathbf{x}} \right\Vert_p \left\Vert {\mathbf{y}} \right\Vert_q$.

Proof

Assume WLOG that $\mathbf x$ and $\mathbf y$ are non-zero.

Let:

$\displaystyle \mathbf u = \langle {u_n} \rangle = \frac {\mathbf x} {\left\Vert {\mathbf x} \right\Vert_p}$

and:

$\displaystyle \mathbf v = \langle {v_n} \rangle = \frac {\mathbf y} {\left\Vert {\mathbf y} \right\Vert_q}$

Then:

$\displaystyle \left\Vert {\mathbf u} \right\Vert_p = \frac {1} {\left\Vert {\mathbf x} \right\Vert_p} \left({ \sum_{n=1}^\infty \left\vert{x_n}\right\vert^p }\right)^{1/p} = 1$

Similarly:

$\left\Vert {\mathbf{v}} \right\Vert_q = 1$

It then suffices to prove that:

$\displaystyle \left\Vert {\mathbf{u}\mathbf{v}} \right\Vert_1 = \frac {\left\Vert {\mathbf{x}\mathbf{y}} \right\Vert_1} {\left\Vert {\mathbf x} \right\Vert_p \left\Vert {\mathbf y} \right\Vert_q} \le 1$

By Young's Inequality for Products:

$\displaystyle \left\vert u_n v_n \right\vert \le \frac 1 p \left\vert u_n \right\vert^p + \frac 1 q \left\vert v_n \right\vert^q$

Summing over all $n \in \N$ gives:

$\displaystyle \left\Vert {\mathbf{u}\mathbf{v}} \right\Vert_1 \le \frac 1 p \left\Vert {\mathbf{u}} \right\Vert_p + \frac 1 q \left\Vert {\mathbf{v}} \right\Vert_q = 1$

as desired.

$\blacksquare$

Also see

Hölder's Inequality helps in proving the Minkowski inequality.

This in turns helps in establishing that $\ell^p$ is a metric space under the metric defined by:

$\displaystyle d \left({x, y}\right) = \left({\sum_k^\infty \left|{x_k - y_k}\right|^p}\right)^{1/p}$

Source of Name

This entry was named for Otto Ludwig Hölder.

It was first found by L. J. Rogers in 1888, and discovered independently by Hölder in 1889.