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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.^[1] A norm is a generalization of the intuitive notion of "length" in the physical world. If ${\displaystyle V}$ is a vector space over ${\displaystyle K}$, where ${\displaystyle K}$ is a field equal to ${\displaystyle \mathbb {R} }$ or to ${\displaystyle \mathbb {C} }$, then a norm on ${\displaystyle V}$ is a map ${\displaystyle V\to \mathbb {R} }$, typically denoted by ${\displaystyle \lVert \cdot \rVert }$, satisfying the following four axioms:

1. Non-negativity: for every ${\displaystyle x\in V}$,${\displaystyle \;\lVert x\rVert \geq 0}$.
2. Positive definiteness: for every ${\displaystyle x\in V}$, ${\displaystyle \;\lVert x\rVert =0}$ if and only if ${\displaystyle x}$ is the zero vector.
3. Absolute homogeneity: for every ${\displaystyle \lambda \in K}$ and ${\displaystyle x\in V}$,
   $${\displaystyle \lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert }$$

   
4. Triangle inequality: for every ${\displaystyle x\in V}$ and ${\displaystyle y\in V}$,
   $${\displaystyle \|x+y\|\leq \|x\|+\|y\|.}$$

   

If ${\displaystyle V}$ is a real or complex vector space as above, and ${\displaystyle \lVert \cdot \rVert }$ is a norm on ${\displaystyle V}$, then the ordered pair ${\displaystyle (V,\lVert \cdot \rVert )}$ 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 ${\displaystyle V}$.

A norm induces a distance, called its (norm) induced metric, 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.

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

for any vectors ${\displaystyle x}$ and ${\displaystyle y.}$

This also shows that a vector norm is a (uniformly) continuous function.

Property 3 depends on a choice of norm ${\displaystyle |\alpha |}$ on the field of scalars. When the scalar field is ${\displaystyle \mathbb {R} }$ (or more generally a subset of ${\displaystyle \mathbb {C} }$), this is usually taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over ${\displaystyle \mathbb {Q} }$ one could take ${\displaystyle |\alpha |}$ to be the ${\displaystyle p}$-adic absolute value.

Topological structure

If ${\displaystyle (V,\|\,\cdot \,\|)}$ is a normed vector space, the norm ${\displaystyle \|\,\cdot \,\|}$ induces a metric (a notion of distance) and therefore a topology on ${\displaystyle V.}$ This metric is defined in the natural way: the distance between two vectors ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$ is given by ${\displaystyle \|\mathbf {u} -\mathbf {v} \|.}$ This topology is precisely the weakest topology which makes ${\displaystyle \|\,\cdot \,\|}$ continuous and which is compatible with the linear structure of ${\displaystyle V}$ in the following sense:

1. The vector addition ${\displaystyle \,+\,:V\times V\to V}$ is jointly continuous with respect to this topology. This follows directly from the triangle inequality.
2. The scalar multiplication ${\displaystyle \,\cdot \,:\mathbb {K} \times V\to V,}$ where ${\displaystyle \mathbb {K} }$ is the underlying scalar field of ${\displaystyle V,}$ is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.

Similarly, for any seminormed vector space we can define the distance between two vectors ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$ as ${\displaystyle \|\mathbf {u} -\mathbf {v} \|.}$ This turns the seminormed space into a pseudometric space (notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence. To put it more abstractly every seminormed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm.

Of special interest are complete normed spaces, which are known as Banach spaces. Every normed vector space ${\displaystyle V}$ sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by ${\displaystyle V}$ and is called the completion of ${\displaystyle V.}$

Two norms on the same vector space are called equivalent if they define the same topology. On a finite-dimensional vector space, all norms are equivalent but this is not true for infinite dimensional vector spaces.

All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same).^[2] And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space ${\displaystyle V}$ is locally compact if and only if the unit ball ${\displaystyle B=\{x:}$Zdroj:https://en.wikipedia.org?pojem=Normed_space
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