Editing Normed vector space

{{short description|Vector space on which a distance is defined}}
{{more footnotes|date=December 2019}}
[[File:Mathematical Spaces.png|thumb|250px|Hierarchy of mathematical spaces. [[Inner product space]]s are a subset of normed vector spaces,
which are a subset of [[metric space]]s, which in turn are a subset of [[topological space]]s.]]

In [[mathematics]], a '''normed vector space''' or '''normed space''' is a [[vector space]] over the [[Real number|real]] or [[Complex number|complex]] numbers on which a [[Norm (mathematics)|norm]] is defined.<ref name="text">{{cite book|first=Frank M.|last=Callier|title=Linear System Theory|location=New York |publisher=Springer-Verlag|year=1991|isbn=0-387-97573-X}}</ref> A norm is a generalization of the intuitive notion of "length" in the physical world. If <math>V</math> is a vector space over <math>K</math>, where <math>K</math> is a field equal to <math>\mathbb R</math> or to <math>\mathbb C</math>, then a norm on <math>V</math> is a map <math>V\to\mathbb R</math>, typically denoted by <math>\lVert\cdot \rVert</math>, satisfying the following four axioms:

#Non-negativity: for every <math>x\in V</math>,<math>\; \lVert x \rVert \ge 0</math>.
#Positive definiteness: for every <math>x \in V</math>, <math>\; \lVert x\rVert=0</math> if and only if <math>x</math> is the zero vector.
# Absolute homogeneity: for every <math>\lambda\in K</math> and <math>x\in V</math>,<math display="block">\lVert \lambda x \rVert = |\lambda|\, \lVert x\rVert </math>
# [[Triangle inequality]]: for every <math>x\in V</math> and <math>y\in V</math>,<math display="block">\|x+y\| \leq \|x\| + \|y\|.</math>

If <math>V</math> is a real or complex vector space as above, and <math>\lVert\cdot\rVert</math> is a norm on <math>V</math>, then the ordered pair <math>(V,\lVert\cdot \rVert)</math> 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 <math>V</math>.

A norm induces a [[Metric (mathematics)|distance]], called its {{em|[[Norm induced metric|(norm) induced metric]]}}, by the formula
<math display="block">d(x,y) = \|y-x\|.</math>
which makes any normed vector space into a [[metric space]] and a [[topological vector space]]. If this metric space is [[Complete metric space|complete]] then the normed space is a <em>[[Banach space]]</em>. 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 sequence]]s 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
<math display=block>d(A, B) = \|\overrightarrow{AB}\|.</math>

The study of normed spaces and Banach spaces is a fundamental part of [[functional analysis]], a major subfield of mathematics.

==Definition==
{{See also|Seminormed space}}

A '''normed vector space''' is a [[vector space]] equipped with a [[Norm (mathematics)|norm]]. A '''{{visible anchor|seminormed vector space}}''' is a vector space equipped with a [[seminorm]].

A useful [[Triangle inequality#Reverse triangle inequality|variation of the triangle inequality]] is
<math display=block>\|x-y\| \geq | \|x\| - \|y\| |</math> 
for any vectors <math>x</math> and <math>y.</math>

This also shows that a vector norm is a ([[Uniform continuity|uniformly]]) [[continuous function]].

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

==Topological structure==

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

#The vector addition <math>\,+\, : V \times V \to V</math> is jointly continuous with respect to this topology. This follows directly from the [[triangle inequality]].
#The scalar multiplication <math>\,\cdot\, : \mathbb{K} \times V \to V,</math> where <math>\mathbb{K}</math> is the underlying scalar field of <math>V,</math> 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 <math>\mathbf{u}</math> and <math>\mathbf{v}</math> as <math>\|\mathbf{u} - \mathbf{v}\|.</math> This turns the seminormed space into a [[pseudometric space]] (notice this is weaker than a metric) and allows the definition of notions such as [[Continuous function (topology)|continuity]] and [[Limit of a function|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 space|complete]] normed spaces, which are known as {{em|[[Banach space]]s}}. 
Every normed vector space <math>V</math> sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by <math>V</math> and is called the {{em|[[Cauchy completion|completion]]}} of <math>V.</math>

Two norms on the same vector space are called {{em|[[Equivalent norm|equivalent]]}} if they define the same [[Topology (structure)|topology]]. On a finite-dimensional vector space (but not infinite-dimensional vector spaces), all norms are equivalent (although the resulting metric spaces need not be the same)<ref>{{Citation|last1=Kedlaya|first1=Kiran S.|author1-link=Kiran Kedlaya|title=''p''-adic differential equations|publisher=[[Cambridge University Press]]|series=Cambridge Studies in Advanced Mathematics|isbn=978-0-521-76879-5|year=2010|volume=125|citeseerx=10.1.1.165.270}}, Theorem 1.3.6</ref> 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 <math>V</math> is [[locally compact]] if and only if the unit ball <math>B = \{ x : \|x\| \leq 1\}</math> is [[Compact space|compact]], which is the case if and only if <math>V</math> is finite-dimensional; this is a consequence of [[Riesz's lemma]].  (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional. The point here is that we don't assume the topology comes from a norm.)

The topology of a seminormed vector space has many nice properties. Given a [[neighbourhood system]] <math>\mathcal{N}(0)</math> around 0 we can construct all other neighbourhood systems as
<math display=block>\mathcal{N}(x) = x + \mathcal{N}(0) := \{x + N : N \in \mathcal{N}(0)\}</math>
with
<math display=block>x + N := \{x + n : n \in N\}.</math>

Moreover, there exists a [[neighbourhood basis]] for the origin consisting of [[Absorbing set|absorbing]] and [[convex set]]s. As this property is very useful in [[functional analysis]], generalizations of normed vector spaces with this property are studied under the name [[locally convex space]]s. 

A norm (or [[seminorm]]) <math>\|\cdot\|</math> on a topological vector space <math>(X, \tau)</math> is continuous if and only if the topology <math>\tau_{\|\cdot\|}</math> that <math>\|\cdot\|</math> induces on <math>X</math> is [[Comparison of topologies|coarser]] than <math>\tau</math> (meaning, <math>\tau_{\|\cdot\|} \subseteq \tau</math>), which happens if and only if there exists some open ball <math>B</math> in <math>(X, \|\cdot\|)</math> (such as maybe <math>\{x \in X : \|x\| < 1\}</math> for example) that is open in <math>(X, \tau)</math> (said different, such that <math>B \in \tau</math>).

== Normable spaces ==

{{See also|Metrizable topological vector space#Normability}}

A [[topological vector space]] <math>(X, \tau)</math> is called '''normable''' if there exists a norm <math>\| \cdot \|</math> on <math>X</math> such that the canonical metric <math>(x, y) \mapsto \|y-x\|</math> induces the topology <math>\tau</math> on <math>X.</math>
The following theorem is due to [[Andrey Kolmogorov|Kolmogorov]]:{{sfn|Schaefer|1999|p=41}}

'''[[Kolmogorov's normability criterion]]''': A Hausdorff topological vector space is normable if and only if there exists a convex, [[von Neumann bounded]] neighborhood of <math>0 \in X.</math>

A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (that is, <math>\neq \{ 0 \}</math>).{{sfn|Schaefer|1999|p=41}} Furthermore, the quotient of a normable space <math>X</math> by a closed vector subspace <math>C</math> is normable, and if in addition <math>X</math>'s topology is given by a norm <math>\|\,\cdot,\|</math> then the map <math>X/C \to \R</math> given by <math display=inline>x + C \mapsto \inf_{c \in C} \|x + c\|</math> is a well defined norm on <math>X / C</math> that induces the [[quotient topology]] on <math>X / C.</math>{{sfn|Schaefer|1999|p=42}}

If <math>X</math> is a Hausdorff [[Locally convex topological vector space|locally convex]] [[topological vector space]] then the following are equivalent:

# <math>X</math> is normable.
# <math>X</math> has a bounded neighborhood of the origin.
# the [[strong dual space]] <math>X^{\prime}_b</math> of <math>X</math> is normable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}
# the strong dual space <math>X^{\prime}_b</math> of <math>X</math> is [[Metrizable topological vector space|metrizable]].{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}

Furthermore, <math>X</math> is finite-dimensional if and only if <math>X^{\prime}_{\sigma}</math> is normable (here <math>X^{\prime}_{\sigma}</math> denotes <math>X^{\prime}</math> endowed with the [[weak-* topology]]).

The topology <math>\tau</math> of the [[Fréchet space]] <math>C^{\infty}(K),</math> as defined in the article on [[spaces of test functions and distributions]], is defined by a countable family of norms but it is {{em|not}} a normable space because there does not exist any norm <math>\|\cdot\|</math> on <math>C^{\infty}(K)</math> such that the topology that this norm induces is equal to <math>\tau.</math> 

Even if a metrizable topological vector space has a topology that is defined by a family of norms, then it may nevertheless still fail to be [[normable space]] (meaning that its topology can not be defined by any {{em|single}} norm). 
An example of such a space is the [[Fréchet space]] <math>C^{\infty}(K),</math> whose definition can be found in the article on [[spaces of test functions and distributions]], because its topology <math>\tau</math> is defined by a countable family of norms but it is {{em|not}} a normable space because there does not exist any norm <math>\|\cdot\|</math> on <math>C^{\infty}(K)</math> such that the topology this norm induces is equal to <math>\tau.</math>  
In fact, the topology of a [[Locally convex topological vector space|locally convex space]] <math>X</math> can be a defined by a family of {{em|norms}} on <math>X</math> if and only if there exists {{em|at least one}} continuous norm on <math>X.</math>{{sfn|Jarchow|1981|p=130}}

==Linear maps and dual spaces==

The most important maps between two normed vector spaces are the [[Continuous function (topology)|continuous]] [[Linear transformation|linear maps]]. Together with these maps, normed vector spaces form a [[Category theory|category]].

The norm is a continuous function on its vector space.  All linear maps between finite-dimensional vector spaces are also continuous.

An ''isometry'' between two normed vector spaces is a linear map <math>f</math> which preserves the norm (meaning <math>\|f(\mathbf{v})\| = \|\mathbf{v}\|</math> for all vectors <math>\mathbf{v}</math>). Isometries are always continuous and [[injective]]. A [[surjective]] isometry between the normed vector spaces <math>V</math> and <math>W</math> is called an ''isometric isomorphism'', and <math>V</math> and <math>W</math> are called ''isometrically isomorphic''. Isometrically isomorphic normed vector spaces are identical for all practical purposes.

When speaking of normed vector spaces, we augment the notion of [[dual space]] to take the norm into account. The dual <math>V^{\prime}</math> of a normed vector space <math>V</math> is the space of all ''continuous'' linear maps from <math>V</math> to the base field (the complexes or the reals) — such linear maps are called "functionals".  The norm of a functional <math>\varphi</math> is defined as the [[supremum]] of <math>|\varphi(\mathbf{v})|</math> where <math>\mathbf{v}</math> ranges over all unit vectors (that is, vectors of norm <math>1</math>) in <math>V.</math> This turns <math>V^{\prime}</math> into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the [[Hahn–Banach theorem]].

==Normed spaces as quotient spaces of seminormed spaces==

The definition of many normed spaces (in particular, [[Banach space]]s) involves a seminorm defined on a vector space and then the normed space is defined as the [[Quotient space (linear algebra)|quotient space]] by the subspace of elements of seminorm zero.  For instance, with the [[Lp space|<math>L^p</math> spaces]], the function defined by
<math display=block>\|f\|_p = \left( \int |f(x)|^p \;dx \right)^{1/p}</math>
is a seminorm on the vector space of all functions on which the [[Lebesgue integral]] on the right hand side is defined and finite.  However, the seminorm is equal to zero for any function [[Support (mathematics)|supported]] on a set of [[Lebesgue measure]] zero.  These functions form a subspace which we "quotient out", making them equivalent to the zero function.

==Finite product spaces==

Given <math>n</math> seminormed spaces <math>\left(X_i, q_i\right)</math> with seminorms <math>q_i : X_i \to \R,</math> denote the [[product space]] by
<math display=block>X := \prod_{i=1}^n X_i</math>
where vector addition defined as
<math display=block>\left(x_1,\ldots,x_n\right) + \left(y_1,\ldots,y_n\right) := \left(x_1 + y_1, \ldots, x_n + y_n\right)</math>
and scalar multiplication defined as
<math display=block>\alpha \left(x_1,\ldots,x_n\right) := \left(\alpha x_1, \ldots, \alpha x_n\right).</math>

Define a new function <math>q : X \to \R</math> by
<math display=block>q\left(x_1,\ldots,x_n\right) := \sum_{i=1}^n q_i\left(x_i\right),</math>
which is a seminorm on <math>X.</math> The function <math>q</math> is a norm if and only if all <math>q_i</math> are norms.

More generally, for each real <math>p \geq 1</math> the map <math>q : X \to \R</math> defined by 
<math display=block>q\left(x_1,\ldots,x_n\right) := \left(\sum_{i=1}^n q_i\left(x_i\right)^p\right)^{\frac{1}{p}}</math>
is a semi norm. 
For each <math>p</math> this defines the same topological space.

A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm.  Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.

== See also ==

* [[Banach space]], normed vector spaces which are complete with respect to the metric induced by the norm
* {{annotated link|Banach–Mazur compactum}}
* [[Finsler manifold]], where the length of each tangent vector is determined by a norm
* [[Inner product space]], normed vector spaces where the norm is given by an [[inner product]]
* {{annotated link|Kolmogorov's normability criterion}}
* [[Locally convex topological vector space]] – a vector space with a topology defined by convex open sets
* [[Space (mathematics)]] – mathematical set with some added structure
* {{annotated link|Topological vector space}}

==References==

{{reflist}}
{{reflist|group=note}}

==Bibliography==
* {{cite book
 | last = Jarchow | first = Hans
 | isbn = 3-519-02224-9
 | mr = 632257
 | publisher = B. G. Teubner, Stuttgart
 | series = Mathematische Leitfäden. [Mathematical Textbooks]
 | title = Locally Convex Spaces
 | year = 1981}}
* {{Rudin Walter Functional Analysis}} <!-- {{sfn|Rudin|1991|pp=}} -->
* {{Banach Théorie des Opérations Linéaires}} <!-- {{sfn|Banach|1932|p=}} -->
* {{Citation|title=Functional analysis and control theory: Linear systems|last=Rolewicz|first=Stefan|year=1987|isbn=90-277-2186-6|publisher=D. Reidel Publishing Co.; PWN—Polish Scientific Publishers|oclc=13064804|edition=Translated from the Polish by Ewa Bednarczuk|series=Mathematics and its Applications (East European Series)|location=Dordrecht; Warsaw|volume=29|pages=xvi+524|mr=920371| doi=10.1007/978-94-015-7758-8}}
* {{cite book|last=Schaefer|first=H. H.|title=Topological Vector Spaces|publisher=Springer New York Imprint Springer|publication-place=New York, NY|year=1999|isbn=978-1-4612-7155-0|oclc=840278135}} <!-- {{sfn|Schaefer|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}}

== External links ==

* {{Commons category-inline|Normed spaces}}

{{Banach spaces}}
{{Functional Analysis}}
{{TopologicalVectorSpaces}}

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[[Category:Normed spaces| ]]