Basic Concepts and Notions in Normed Spaces

Vector space on which a distance is defined.

Normed spaces are a fundamental concept in the field of functional analysis and they play a significant role in various branches of mathematics, physics, and engineering. This article will introduce the basic concepts and notions related to normed spaces.

Definition of Normed Spaces

A normed space is a pair (V, ||.||), where V is a vector space over the field F (which is either the field of real numbers R or the field of complex numbers C) and ||.|| is a function from V to R (the set of non-negative real numbers) that satisfies the following conditions for all vectors x, y in V and all scalars α in F:

1. ||x|| ≥ 0 (Non-negativity)
2. ||x|| = 0 if and only if x = 0 (Definiteness)
3. ||αx|| = |α| ||x|| (Homogeneity or scalability)
4. ||x + y|| ≤ ||x|| + ||y|| (Triangle inequality)

The function ||.|| is called a norm on V, and the vector space V equipped with this norm is called a normed space.

Examples of Normed Spaces

1. The set of real numbers R with the absolute value function as the norm is a normed space.
2. The set of n-dimensional real or complex vectors (R^n or C^n) with the Euclidean norm (also known as the 2-norm or the L2 norm) is a normed space.
3. The set of all continuous real-valued functions defined on a closed interval [a, b] with the supremum norm (also known as the infinity norm or the L∞ norm) is a normed space.

Norms on Finite-Dimensional Spaces

In a finite-dimensional space, all norms are equivalent. This means that given any two norms ||.||1 and ||.||2 on a finite-dimensional vector space V, there exist positive real numbers C1 and C2 such that for all x in V, we have:

C1 ||x||1 ≤ ||x||2 ≤ C2 ||x||1

Properties of Norms

Norms in a normed space have several important properties that are direct consequences of the definition:

1. The zero vector has zero norm.
2. The norm of a vector is the same as the norm of its negative.
3. The norm of a sum of two vectors is less than or equal to the sum of their norms (Triangle Inequality).
4. The norm of a scalar multiple of a vector is the absolute value of the scalar times the norm of the vector.

In conclusion, normed spaces are a central object of study in functional analysis. Understanding the basic concepts and properties of normed spaces is crucial for further study in this area.