Definition:P-Norm

Definition

Let $p \ge 1$ be a real number.

Let $\BB$ be a Banach space.

Let $\ell^p$ denote the $p$-sequence space in $\BB$:

$\ds \ell^p := \set {\sequence {s_n}_{n \mathop \in \N} \in \BB^\N: \sum_{n \mathop = 0}^\infty \norm {s_n}^p < \infty}$

Let $\mathbf s = \sequence {s_n} \in \ell^p$ be a sequence in $\ell^p$.

Then the $p$-norm of $\mathbf s$ is defined as:

$\ds \norm {\mathbf s}_p = \paren {\sum_{n \mathop = 0}^\infty \size {s_n}^p}^{1 / p}$

This is often presented in expository treatments either on the real number line or the complex plane:

Real Number Line

Let ${\ell^p}_\R$ denote the real $p$-sequence space:

$\ds {\ell^p}_\R := \set {\sequence {x_n}_{n \mathop \in \N} \in \R^\N: \sum_{n \mathop = 0}^\infty \size {x_n}^p < \infty}$

Let $\mathbf x = \sequence {x_n} \in {\ell^p}_\R$ be a sequence in ${\ell^p}_\R$.

Then the $p$-norm of $\mathbf x$ is defined as:

$\ds \norm {\mathbf x}_p = \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^p}^{1 / p}$

Complex Plane

Let ${\ell^p}_\C$ denote the complex $p$-sequence space:

$\ds {\ell^p}_\C := \set {\sequence {z_n}_{n \mathop \in \N} \in \C^\N: \sum_{n \mathop = 0}^\infty \cmod {z_n}^p < \infty}$

Let $\mathbf z = \sequence {z_n} \in {\ell^p}_\C$ be a sequence in ${\ell^p}_\C$.

Then the $p$-norm of $\mathbf z$ is defined as:

$\ds \norm {\mathbf z}_p = \paren {\sum_{n \mathop = 0}^\infty \cmod {z_n}^p}^{1 / p}$

Also see

• $p$-Norm is Norm
• Derivative of $p$-Norm wrt $p$
• $p$-Norm of Real Sequence is Strictly Decreasing Function of $p$
• Transformation of $p$-Norm

• Definition:Euclidean Norm: for $p = 2$, that is, the $2$-norm

• Definition:Taxicab Norm: for $p = 1$, that is, the $1$-norm

• Results about $p$-norms can be found here.

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• By the triangle inequality, the $1$-norm satisfies the Norm Axiom $\text N 3$: Triangle Inequality.

• For $p > 1$, Minkowski's Inequality for Sums states that the $p$-norm satisfies the Norm Axiom $\text N 3$: Triangle Inequality.

• The $p$-norm is not to be confused with the $p$-adic norm.

Sources

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• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): $p$-norm