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.
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:
The function ||.|| is called a norm on V, and the vector space V equipped with this norm is called a normed space.
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
Norms in a normed space have several important properties that are direct consequences of the definition:
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.
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