Category:Hölder's Inequality for Sums
This category contains pages concerning 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:
- $\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$
Pages in category "Hölder's Inequality for Sums"
The following 12 pages are in this category, out of 12 total.
H
- Hölder's Inequality for Finite Sums
- Hölder's Inequality for Finite Sums/Euclidean Plane
- Hölder's Inequality for Sums
- Hölder's Inequality for Sums/Also known as
- Hölder's Inequality for Sums/Equality
- Hölder's Inequality for Sums/Finite
- Hölder's Inequality for Sums/Finite/Proof
- Hölder's Inequality for Sums/Formulation 1
- Hölder's Inequality for Sums/Formulation 1/Equality
- Hölder's Inequality for Sums/Formulation 2
- Hölder's Inequality for Sums/Formulation 2/Equality
- Hölder's Inequality for Sums/Parameter Inequalities