Hölder's Inequality for Sums/Formulation 1/Equality

Theorem

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

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

Let:

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

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

where $\ell^p$ denotes the $p$-sequence space.

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

Hölder's Inequality for Sums states that:

$\mathbf x \mathbf y \in \ell^1$

and:

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

We have that:

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

if and only if:

$\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$

for some real constant $c$.

Proof

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