Editing Normed vector space (section)

==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>).