In mathematics, a normed vector space or normed space is a vector space, typically 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 V {\displaystyle V} is a vector space over K {\displaystyle K} , where K {\displaystyle K} is a field equal to R {\displaystyle \mathbb {R} } or to C {\displaystyle \mathbb {C} } , then a norm on V {\displaystyle V} is a map V → R {\displaystyle V\to \mathbb {R} } , typically denoted by ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } , satisfying the following four axioms:
Non-negativity: for every x ∈ V {\displaystyle x\in V} , ‖ x ‖ ≥ 0 {\displaystyle \;\lVert x\rVert \geq 0} . Positive definiteness: for every x ∈ V {\displaystyle x\in V} , ‖ x ‖ = 0 {\displaystyle \;\lVert x\rVert =0} if and only if x {\displaystyle x} is the zero vector. Absolute homogeneity: for every λ ∈ K {\displaystyle \lambda \in K} and x ∈ V {\displaystyle x\in V} , ‖ λ x ‖ = | λ | ‖ x ‖ {\displaystyle \lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert }
Triangle inequality: for every x ∈ V {\displaystyle x\in V} and y ∈ V {\displaystyle y\in V} , ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ . {\displaystyle \|x+y\|\leq \|x\|+\|y\|.}
If V {\displaystyle V} is a real or complex vector space as above, and ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } is a norm on V {\displaystyle V} , then the ordered pair ( V , ‖ ⋅ ‖ ) {\displaystyle (V,\lVert \cdot \rVert )} 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 V {\displaystyle V} . A norm induces a distance, called its (norm) induced metric, by the formula
d ( x , y ) = ‖ y − x ‖ . {\displaystyle d(x,y)=\|y-x\|.}
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 the 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
d ( A , B ) = ‖ A B → ‖ . {\displaystyle d(A,B)=\|{\overrightarrow {AB}}\|.}
The study of normed spaces and Banach spaces is a fundamental part of functional analysis, a major subfield of mathematics.
Definition
A normed vector space is a vector space equipped with a norm. A seminormed vector space is a vector space equipped with a seminorm. A useful variation of the triangle inequality is
‖ x − y ‖ ≥ | ‖ x ‖ − ‖ y ‖ | {\displaystyle \|x-y\|\geq |\|x\|-\|y\||} for any vectors x {\displaystyle x} and y . {\displaystyle y.}
This also shows that a vector norm is a (uniformly) continuous function. Property 3 depends on a choice of norm | α | {\displaystyle |\alpha |} on the field of scalars. When the scalar field is R {\displaystyle \mathbb {R} } (or more generally a subset of C {\displaystyle \mathbb {C} } ), this is usually taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over Q {\displaystyle \mathbb {Q} } one could take | α | {\displaystyle |\alpha |} to be the p {\displaystyle p} -adic absolute value.
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