Hölder's Inequality for Sums/Equality
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Theorem
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 $\mathbf x$.
Formulation $1$: Condition for Equality
- $\norm {\mathbf x \mathbf y}_1 = \norm {\mathbf x}_p \norm {\mathbf y}_q$
- $\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
Formulation $2$
Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be (possibly infinite) convergent sequences in $\GF$.
Formulation $2$: Condition for Equality
- $\ds \sum_{k \mathop \in \N} \size {x_k y_k} = \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$
- $\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$