NORMED VECTOR SPACE Meaning and
Definition
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A normed vector space is a mathematical concept that combines two fundamental algebraic structures: a vector space and a norm. A vector space is a set of objects, called vectors, together with two operations: vector addition and scalar multiplication. These operations satisfy certain properties such as commutativity, associativity, and distributivity. Vectors can be thought of as elements in an n-dimensional space, where n is a positive integer.
In a normed vector space, in addition to the vector space structure, there is also a norm defined. A norm is a real-valued function that assigns a length or magnitude to each vector in the space. This function satisfies properties such as non-negativity, homogeneity, and the triangle inequality. The norm provides a measure of distance or size for vectors and is often represented by double vertical bars: ||x||.
The combination of a vector space and a norm in a normed vector space introduces the concept of distance and convergence. The norm allows us to define the distance between two vectors as the length of their difference. This notion of distance enables the study of convergence, limits, and continuity in the normed vector space.
Normed vector spaces find applications in various branches of mathematics, such as functional analysis, linear algebra, and numerical analysis. They provide a framework for analyzing the behavior of vectors and their operations in a quantitative sense, allowing for the development of rigorous mathematical theories and applications in a wide range of areas.
