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Concept
Normed vector space
Summary
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms:
Non-negativity: for every ,.
Positive definiteness: for every , if and only if is the zero vector.
Absolute homogeneity: for every and ,
Triangle inequality: for every and ,
If is a real or complex vector space as above, and is a norm on , then the ordered pair is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by .
A norm induces a distance, called its , by the formula
which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm.
An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance by the formula
The study of normed spaces and Banach spaces is a fundamental part of functional analysis, a major subfield of mathematics.
Seminormed space
A normed vector space is a vector space equipped with a norm. A is a vector space equipped with a seminorm.
A useful variation of the triangle inequality is
for any vectors and
This also shows that a vector norm is a (uniformly) continuous function.
Property 3 depends on a choice of norm on the field of scalars.
Official source
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