nLab
p-norm

Contents

Idea

For pp \in \mathbb{R}, p1p \geq 1, the pp-norm is a norm on suitable real vector spaces given by the ppth root of the sum (or integral) of the ppth-powers of the absolute values of the vector components. With due care the definition makes sense for non-finite dimensional vector spaces such as sequence spaces and Lebesgue spaces, making them into normed vector spaces, hence metric spaces.

For p=2p = 2 the pp-norm is the standard Euclidean norm, defining Euclidean spaces and Hilbert spaces of square integrable functions.

For p=p = \infty the pp-norm (found by taking the limit pp \to \infty) is the supremum (or essential supremum in the continuous case) of the absolute values of the components of vectors, then called the supremum norm.

For 0p<10 \leq p \lt 1 one may still make sense of the formulas that define pp-norms for p1p \geq 1 (see at Generalizations below), but the resulting concepts are no longer genuine norms.

Definition

The concept of pp-norm makes sense in increasing generality,

The pp-norm on finite dimensional vector spaces

For nn \in \mathbb{N}, pp \in \mathbb{R}, p>0p \gt 0, the pp-norm p\Vert - \Vert_p is the norm on the real finite dimensional vector space n\mathbb{R}^n given by the ppth root of the sum of the pp-powers of the absolute value of the components of a given vector x=(x) i=1 n n\vec x = (x)_{i = 1}^n \in \mathbb{R}^n:

x p i|x i| pp {\Vert \vec x \Vert_p} \coloneqq \root p {\sum_i {\vert x_i\vert^p}}

Equipping it with this norm makes n\mathbb{R}^n a normed vector space.

For p=2p = 2 this is the Euclidean norm, the standard norm that defines Euclidean space.

For p=p = \infty one takes the supremum over the absolute values of the components

x sup1in|x i|. {\Vert \vec x\Vert_\infty} \;\coloneqq\; \underset{1 \leq i \leq n}{sup} {\vert x_i\vert} \,.

The graphics on the right (grabbed from Wikipedia) shows unit circles in 2\mathbb{R}^2 with respect to various p-norms.

The 𝓁 p\mathcal{l}^p-norm on sequence spaces

For pp \in \mathbb{R}, write p\ell^p for the vector space of those sequences (x i) i(x_i)_{i \in \mathbb{N}} in \mathbb{R} for which the series

i|x i| p< \underset{i \in \mathbb{N}}{\sum} {\vert x_i\vert}^p \;\lt\; \infty

(the sum of the ppth powers of the absolute value of the components of the sequence) converges.

For p1p \geq 1 the the function

p: p {\Vert -\Vert_p} \;\colon\; \ell^p \longrightarrow \mathbb{R}
(x i) i pi|x i| pp {\Vert (x_i)_{i \in \mathbb{N}} \Vert_p} \;\coloneqq\; \root{p}{\underset{i \in \mathbb{N}}{\sum} {\vert x_i\vert^p}}

defines a norm on this real vector space. This normed vector space is complete, hence a Banach space. This is called the sequence space.

For p=p = \infty one takes \ell^\infty to be the space of bounded sequences and

(x i) i supk|x k| {\Vert (x_i)_{i \in \mathbb{N}} \Vert_\infty} \;\coloneqq\; \underset{k \in \mathbb{N}}{sup} {\vert x_k\vert }

to be the supremum over the absolute values of the components of the sequence. This is also called the supremum norm.

The L pL^p-norm on Lebesgue spaces

More generally, for (X,μ)(X,\mu) a measure space, write L p(X)L^p(X) for the vector space of equivalence classes of those measurable functions f:Xf \colon X \to \mathbb{R}, for which the integral

X|f| pdμ< \int_X {\vert f \vert^p} d\mu \;\lt\; \infty

exists, and where two such functions are regarded as equivalent, f 1f 2f_1 \sim f_2, if

X|f 2f 1| pdμ=0. \int_X {\vert f_2 - f_1 \vert^p} d\mu \;=\; 0 \,.

On this space the function

p:L p(X) {\Vert -\Vert_p} \;\colon\; L^p(X) \longrightarrow \mathbb{R}
f p X|f| ppdμ {\Vert f \Vert_p} \;\coloneqq\; \root{p}{\int_X {\vert f\vert^p}} d\mu

defines a norm. The triangle inequality holds due to Minkowski's inequality. The normed vector space (L p(X), p)(L^p(X), {\Vert- \Vert_p}) is also called a Lebesgue space.

Generalizations for 0p<10 \leq p \lt 1

For 0p<10 \leq p \lt 1, the above definitions for p\Vert {-}\Vert_p still make sense in themselves, but the result is no longer a norm, as Minkowski's inequality (the triangle inequality for pp-norms) fails.

A variant definition for 0<p10 \lt p \leq 1 (which agrees with the usual definition for p=1p = 1, preserving continuity in pp) leaves out the ppth root; then the result satisfies the triangle inequality (and indeed is a metric) but fails to be a norm because it is not positive-homogeneous of degree 11 (but of degree pp instead). Such a thing is called an F-norm.

For p=0p = 0, we might try to take the limit as p0p \searrow 0. For the unmodified pp-norm (with the root), this is infinite if there is more than one nonzero entry and is the absolute value of the one nonzero entry if there is only one (or 0 if there is none); for the modified pp-norm (without the root), it is the (possibly infinite) number of nonzero entries. In either case, however the triangle inequality fails. Therefore, there is a further modified 00-norm, given by

(x 1,x 2,) 0= n=1 2 n|x n|1+|x n| {\|(x_1, x_2, \ldots)\|_0} = \sum_{n=1}^\infty \frac {2^{-n} {|x_n|}} {1 + {|x_n|}}

for l 0l^0, and this is an FF-norm. (But I don't know what is the justification for thinking of this as a pp-norm for p=0p = 0.)

References

Last revised on July 5, 2017 at 04:50:37. See the history of this page for a list of all contributions to it.