Normed Spaces

Norm

Norm

A norm on a vector space is a map such that

1. if and only if .
2. for every and , we call this homogeneity .
3. for every , we call this triangle inequality.

If is a vector space and is a norm on , then the pair is called a normed space.

e.g. We define the vector space of real-valued continuous functions on the interval . The standard norm to use on is the supremum norm: Another family of norms are defined on using an integral: for set And we can also define the inner product:

Proposition

for all for norm .

Proposition

If is a normed space and is a subspace of , then is also a normed space.

Euclidean Norm

For , we define the Euclidean () norm as

Norm

The norm of is defined as Specifically, we have and . Specially, we define the norm as the number of non-zero elements in a vector, and as

Minkowski’s Inequality

In , for all , if then

Sets from Norms

Def Closed Unit Ball The closed unit ball in a normed space is the set

Lemma

In any normed space , the closed unit ball is convex.

Proof For all , we have Hence .

Lemma

Suppose a function satisfies first and second properties of the definition of a norm and the set is convex then satisfies the triangle inequality: and so defines a norm on .

Proof Clearly if then we have , satisfies the triangle inequality. So we can assume that and . As is convex, we have So , it follows that Hence by property of function :

Equivalence Relation of Norms

Comparable Norms

Two norms and on vector space are comparable or equivalent if there exists constants such that It is easy to check that this is an equivalence relation.

Proposition

Two norms are equivalent if and only there exist constants such that where is the closed unit ball in with .