Norm

An understanding of a norm is needed to proceed to linear isometries

Normed vector spaces

A normed vector space is a vector space equipped with a norm

$$\|\cdot\|_V$$ , it may be denoted $$(V,\|\cdot\|_V,F)$$

Definition

A norm on a vector space $(V,F)$ is a function

$$\|\cdot\|:V\rightarrow\mathbb{R}$$ such that:

1. $$\forall x\in V\ \|x\|\ge 0$$
2. $$\|x\|=0\iff x=0$$
3. $$\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|$$ where $$|\cdot|$$ denotes absolute value
4. $$\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|$$ - a form of the triangle inequality

Often parts 1 and 2 are combined into the statement

• $$\|x\|\ge 0\text{ and }\|x\|=0\iff x=0$$ so only 3 requirements will be stated.

I don't like this

Norms may define a metric space

To get a metric space from a norm simply define

$$d(x,y)=\|x-y\|$$

Common norms

Name	Norm	Notes
Norms on $$\mathbb{R}^n$$
1-norm	$$\|x\|_1=\sum^n_{i=1}|x_i|$$	it's just a special case of the p-norm.
2-norm	$$\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}$$	Also known as the Euclidean norm (see below) - it's just a special case of the p-norm.
p-norm	$$\|x\|_p=\left(\sum^n_{i=1}|x_i|^p\right)^\frac{1}{p}$$	(I use this notation because it can be easy to forget the $$p$$ in $$\sqrt[p]{}$$ )
$$\infty-$$ norm	$$\|x\|_\infty=\sup(\{x_i\}_{i=1}^n)$$	Also called $$\infty-$$ norm
Norms on $$\mathcal{C}([0,1],\mathbb{R})$$
$$\|\cdot\|_{L^p}$$	$$\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)^\frac{1}{p}$$	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.
Induced norms
Pullback norm	$$\|\cdot\|_U$$	For a linear isomorphism $$L:U\rightarrow V$$ where V is a normed vector space

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

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TODO: proof

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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

• Euclidean norm