What are some advantages and disadvantages of working with metric spaces versus normed spaces?

1 What is a metric space?

A metric space is a set of elements, called points, together with a function, called a metric, that assigns a non-negative number to any pair of points. The metric satisfies some properties that make it behave like a distance: it is zero if and only if the points are the same, it is symmetric, and it obeys the triangle inequality. A metric space can be very general and abstract, and it does not require any additional structure or operations on the points.

2 What is a normed space?

A normed space is a special kind of metric space, where the points are also vectors, and the metric is derived from a function, called a norm, that assigns a non-negative number to any vector. The norm satisfies some properties that make it behave like a size or a length: it is zero if and only if the vector is zero, it is positive for any non-zero vector, it is homogeneous, and it obeys the triangle inequality. A normed space also has a vector addition and a scalar multiplication, which make it a linear space.

3 Advantages of metric spaces

One of the main advantages of metric spaces is their generality and flexibility. You can define a metric on any set of objects, as long as it satisfies the distance properties. For example, you can define a metric on words, images, graphs, or probability distributions. This allows you to study various types of functions and problems that may not fit into a linear or normed framework. Another advantage of metric spaces is that they often have simpler and more intuitive definitions and proofs, since they do not involve any extra operations or structures.

4 Disadvantages of metric spaces

One of the main disadvantages of metric spaces is their lack of algebraic structure and properties. Since metric spaces do not have any vector operations, you cannot perform linear combinations, scalar products, or matrix multiplications on them. This limits the tools and techniques that you can use to analyze and manipulate functions and operators on metric spaces. Another disadvantage of metric spaces is that they may be too general and abstract, and lose some information or features that are relevant for specific problems or applications.

5 Advantages of normed spaces

One of the main advantages of normed spaces is their rich algebraic structure and properties. Since normed spaces are also linear spaces, you can use vector operations, scalar products, matrix multiplications, and other linear algebra tools and techniques to analyze and manipulate functions and operators on normed spaces. This opens up many possibilities and results that are not available for general metric spaces. Another advantage of normed spaces is that they often have more concrete and specific definitions and proofs, since they involve additional operations and structures.

6 Disadvantages of normed spaces

One of the main disadvantages of normed spaces is their restriction and rigidity. You cannot define a norm on any set of objects, as it has to satisfy the size properties and be compatible with the vector operations. For example, you cannot define a norm on words, images, graphs, or probability distributions, unless you embed them into some linear space. This may limit the types of functions and problems that you can study in a normed framework. Another disadvantage of normed spaces is that they may be too complex and technical, and obscure some intuition or simplicity that are present in general metric spaces.