Properties of Normed Spaces

Vector space on which a distance is defined.

A normed space is a central object of study in functional analysis. It is a vector space with a norm that allows for the definition of length and angle. In this unit, we will explore the properties of normed spaces, including their structure as metric spaces, the concepts of open and closed sets, bounded sets, and convergence.

Normed Spaces as Metric Spaces

A normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is a function that assigns a strictly positive length or size to each vector in the vector space—except for the zero vector, which is assigned a length of zero.

A normed space is a metric space, with the metric (or distance function) induced by the norm. The distance between two vectors u and v in a normed space is given by the norm of their difference ||u-v||.

Open Sets, Closed Sets, and Closure in Normed Spaces

In a normed space, we can define open and closed sets using the metric induced by the norm. An open set is a set where, for every point in the set, there exists some positive real number such that all points within that distance from the given point are also in the set.

A set is closed if its complement is open. In other words, a set is closed if it contains all its limit points.

The closure of a set is the smallest closed set that contains the set. It can be found by adding all the limit points of the set to the set itself.

Bounded Sets in Normed Spaces

A set in a normed space is bounded if there exists some real number M such that the norm of every vector in the set is less than M. In other words, a set is bounded if it can be contained within a ball of finite radius.

Convergence and Cauchy Sequences in Normed Spaces

A sequence in a normed space is said to converge if, as the index goes to infinity, the elements of the sequence get arbitrarily close to a certain point. This point is called the limit of the sequence.

A sequence is a Cauchy sequence if, for every positive real number, there exists a positive integer N such that the distance between any two elements of the sequence with indices greater than N is less than the given real number.

In a normed space, every convergent sequence is a Cauchy sequence, but the converse is not always true. However, in a complete normed space (or Banach space), every Cauchy sequence is convergent.

In conclusion, normed spaces provide a rich structure that allows us to generalize many concepts from Euclidean space to more abstract settings. This makes them a powerful tool in many areas of mathematics and physics.