Hölder's Inequality for Sums

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


The proof of Hölder's inequality involves establishing an auxiliary inequality in the first stage and then using that auxiliary inequality to prove Hölder's inequality.

Let $p > 1$ and choose $q$ to be such that $\dfrac 1 p + \dfrac 1 q = 1$.

Hence $\left({p-1}\right) \left({q-1}\right) = 1$ and so $\dfrac 1 {p-1} = q-1$.

Accordingly $u = t^{p-1} \iff t = u^{q-1}$.

Let $\alpha, \beta$ be any positive real numbers.

Since $\alpha \beta$ is the area of the rectangle in the given figure, we have:


 * $\displaystyle \alpha \beta \le \int_0^\alpha t^{p-1} \ \mathrm d t + \int_0^\beta u^{q-1} \ \mathrm d u = \frac {\alpha^p} p + \frac {\beta^q} q$

Note that even if the graph intersected the side of the rectangle corresponding to $t = \alpha$, this inequality would hold.

Also note that if either of $\alpha, \beta$ were zero then this inequality would hold trivially.

Now we turn our attention towards proving Hölder's Inequality.

Without loss of generality, assume 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:
 * $\displaystyle \mathbf{v} = \langle {v_n} \rangle = \frac {\mathbf{y}} {\left\Vert {\mathbf{y}} \right\Vert_q}$

Clearly:
 * $\left\Vert {\mathbf{u}} \right\Vert_p = \left\Vert {\mathbf{v}} \right\Vert_q = 1$

It then suffices to prove that:
 * $\left\Vert {\mathbf{u}\mathbf{v}} \right\Vert_1 \le 1$

By the inequality proved above:
 * $\displaystyle \left\vert u_n v_n \right\vert \le \frac 1 p \left\vert u_n \right\vert^p + \frac 1 q \left\vert v_n \right\vert^q$

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.

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

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