Definition:P-Norm/Real

Definition

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

Let $\R$ denote the 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:

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

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.