# What is the difference between normed space and metric space?

## What is the difference between normed space and metric space?

While a metric provides us with a notion of the distance between points in a space, a norm gives us a notion of the length of an individual vector. A norm can only be defined on a vector space, while a metric can be defined on any set.

**What is meant by normed space?**

Definition. A normed space is a vector space X endowed with a function X→[0,∞),x↦‖x‖, called the norm on X, which satisfies: (i)‖λx‖=|λ|‖x‖,(positive homogeneity)(ii)‖x+y‖≤‖x‖+‖y‖,(triangle inequality)(iii)‖x‖=0if and only if x=0,(positive definiteness) for all scalars λ and all elements x,y∈X.

### Is normed vector space bounded?

Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.

**How do norm and metric differ?**

A metric measures distances between pairs of things. A norm measures the size of a single thing.

## Which is not a normed vector or metric space?

Let Cbe the unit circle fx2V jjjxjj= 1g. This is another example of a metric space that is not a normed vector space: V is a metric space, using the metric de\\fned from jjjj, and therefore, according to the above remark, so is C; but Cis not a vector space, so it is not a normed vector space.

**Are there metrics that do not arise as norms?**

The following metrics do not arise as norms [otherwise we must have d (a x, a y) = |a| d ( x, y )]. Defn A normed linear space is a vector space X and a non-negative valued mapping ||. || on X, called the norm , which satisfies the properties ||x||=0 if and only if x=0. ||a x|| =|a| ||x||, for all scalars a.

### How to prove that a metric space is a linear space?

We prove that each of the above are metric spaces by showing that they are normed linear spaces, where the obvious candidates are used for norms. The following metrics do not arise as norms [otherwise we must have d (a x, a y) = |a| d ( x, y )].

**Is the pair ( x ; d ) a metric space?**

The pair (X;d) is called a metric space. Remark: If jjjjis a norm on a vector space V, then the function d: V V !R. + de ned by d(x;x0) := jjx x0jjis a metric on V In other words, a normed vector space is automatically a metric space, by de ning the metric in terms of the norm in the natural way.