Difference between revisions of "Norm"
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==Common norms== | ==Common norms== | ||
| − | === | + | {| class="wikitable" border="1" |
| − | <math>\|x\|_1=\sum^n_{i=1}|x_i|</math> | + | |- |
| − | + | ! Name | |
| − | <math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math> | + | ! Norm |
| − | + | ! Notes | |
| − | <math>\|x\|_p=\left(\sum^n_{i=1}|x_i|^p\right)^\frac{1}{p}</math> (I use this notation because it can be easy to forget the <math>p</math> in <math>\sqrt[p]{}</math>) | + | |- |
| − | + | !colspan="3"|Norms on <math>\mathbb{R}^n</math> | |
| − | Also called <math>\infty-</math>norm<br/> | + | |- |
| − | <math>\|x\|_\infty=\ | + | | 1-norm |
| + | |<math>\|x\|_1=\sum^n_{i=1}|x_i|</math> | ||
| + | |it's just a special case of the p-norm. | ||
| + | |- | ||
| + | | 2-norm | ||
| + | |<math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math> | ||
| + | | Also known as the Euclidean norm (see below) - it's just a special case of the p-norm. | ||
| + | |- | ||
| + | | p-norm | ||
| + | |<math>\|x\|_p=\left(\sum^n_{i=1}|x_i|^p\right)^\frac{1}{p}</math> | ||
| + | |(I use this notation because it can be easy to forget the <math>p</math> in <math>\sqrt[p]{}</math>) | ||
| + | |- | ||
| + | | <math>\infty-</math>norm | ||
| + | |<math>\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)</math> | ||
| + | |Also called <math>\infty-</math>norm<br/> | ||
| + | |- | ||
| + | !colspan="3"|Norms on <math>\mathcal{C}([0,1],\mathbb{R})</math> | ||
| + | |- | ||
| + | | <math>\|\cdot\|_{L^p}</math> | ||
| + | | <math>\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)</math> | ||
| + | | '''NOTE''' be careful extending to interval <math>[a,b]</math> as proof it is a norm relies on having a unit measure | ||
| + | |- | ||
| + | | <math>\infty-</math>norm | ||
| + | | <math>\|f\|_\infty=\sup_{x\in[0,1]}(|f(x)|)</math> | ||
| + | | Following the same spirit as the <math>\infty-</math>norm on <math>\mathbb{R}^n</math> | ||
| + | |- | ||
| + | | <math>\|\cdot\|_{C^k}</math> | ||
| + | | <math>\|f\|_{C^k}=\sum^k_{i=1}\sup_{x\in[0,1]}(|f^{(i)}|)</math> | ||
| + | | here <math>f^{(k)}</math> denotes the <math>k^\text{th}</math> derivative. | ||
| + | |} | ||
==Equivalence of norms== | ==Equivalence of norms== | ||
Revision as of 03:17, 8 March 2015
Contents
[hide]Definition
A norm on a vector space (V,F) is a function ∥⋅∥:V→R such that:
- ∀x∈V ∥x∥≥0
- ∥x∥=0⟺x=0
- ∀λ∈F,x∈V ∥λx∥=|λ|∥x∥ where |⋅| denotes absolute value
- ∀x,y∈V ∥x+y∥≤∥x∥+∥y∥ - a form of the triangle inequality
Often parts 1 and 2 are combined into the statement
- ∥x∥≥0 and ∥x∥=0⟺x=0 so only 3 requirements will be stated.
I don't like this
Common norms
| Name | Norm | Notes |
|---|---|---|
| Norms on Rn | ||
| 1-norm | ∥x∥1=n∑i=1|xi| | it's just a special case of the p-norm. |
| 2-norm | ∥x∥2=√n∑i=1x2i | Also known as the Euclidean norm (see below) - it's just a special case of the p-norm. |
| p-norm | ∥x∥p=(n∑i=1|xi|p)1p | (I use this notation because it can be easy to forget the p in p√) |
| ∞−norm | ∥x∥∞=sup | Also called \infty-norm |
| Norms on \mathcal{C}([0,1],\mathbb{R}) | ||
| \|\cdot\|_{L^p} | \|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right) | NOTE be careful extending to interval [a,b] as proof it is a norm relies on having a unit measure |
| \infty-norm | \|f\|_\infty=\sup_{x\in[0,1]}(|f(x)|) | Following the same spirit as the \infty-norm on \mathbb{R}^n |
| \|\cdot\|_{C^k} | \|f\|_{C^k}=\sum^k_{i=1}\sup_{x\in[0,1]}(|f^{(i)}|) | here f^{(k)} denotes the k^\text{th} derivative. |
Equivalence of norms
Given two norms \|\cdot\|_1 and \|\cdot\|_2 on a vector space V we say they are equivalent if:
\exists c,C\in\mathbb{R}\ \forall x\in V:\ c\|x\|_1\le\|x\|_2\le C\|x\|_1
We may write this as \|\cdot\|_1\sim\|\cdot\|_2 - this is an Equivalence relation
TODO: proof
Examples
- Any two norms on \mathbb{R}^n are equivalent
- The norms \|\cdot\|_{L^1} and \|\cdot\|_\infty on \mathcal{C}([0,1],\mathbb{R}) are not equivalent.
Examples
The Euclidean Norm
TODO: Migrate this norm to its own page
The Euclidean norm is denoted \|\cdot\|_2
Here for x\in\mathbb{R}^n we have:
\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}
Proof that it is a norm
TODO: proof
Part 4 - Triangle inequality
Let x,y\in\mathbb{R}^n
\|x+y\|_2^2=\sum^n_{i=1}(x_i+y_i)^2 =\sum^n_{i=1}x_i^2+2\sum^n_{i=1}x_iy_i+\sum^n_{i=1}y_i^2 \le\sum^n_{i=1}x_i^2+2\sqrt{\sum^n_{i=1}x_i^2}\sqrt{\sum^n_{i=1}y_i^2}+\sum^n_{i=1}y_i^2 using the Cauchy-Schwarz inequality
=\left(\sqrt{\sum^n_{i=1}x_i^2}+\sqrt{\sum^n_{i=1}y_i^2}\right)^2 =\left(\|x\|_2+\|y\|_2\right)^2
Thus we see: \|x+y\|_2^2\le\left(\|x\|_2+\|y\|_2\right)^2, as norms are always \ge 0 we see:
\|x+y\|_2\le\|x\|_2+\|y\|_2 - as required.
