Editing Banach space (section)

====Complete norms vs complete metrics====

A metric <math>D</math> on a vector space <math>X</math> is induced by a norm on <math>X</math> if and only if <math>D</math> is [[translation invariant]]<ref group=note name="translation invariant metric"/> and ''absolutely homogeneous'', which means that <math>D(sx, sy) = |s| D(x, y)</math> for all scalars <math>s</math> and all <math>x, y \in X,</math> in which case the function <math>\|x\| := D(x, 0)</math> defines a norm on <math>X</math> and the canonical metric induced by <math>\|{\cdot}\|</math> is equal to <math>D.</math>

Suppose that <math>(X, \|{\cdot}\|)</math> is a normed space and that <math>\tau</math> is the norm topology induced on <math>X.</math> Suppose that <math>D</math> is {{em|any}} [[Metric (mathematics)|metric]] on <math>X</math> such that the topology that <math>D</math> induces on <math>X</math> is equal to <math>\tau.</math> If <math>D</math> is [[translation invariant]]<ref group=note name="translation invariant metric"/> then <math>(X, \|{\cdot}\|)</math> is a Banach space if and only if <math>(X, D)</math> is a complete metric space.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
If <math>D</math> is {{em|not}} translation invariant, then it may be possible for <math>(X, \|{\cdot}\|)</math> to be a Banach space but for <math>(X, D)</math> to {{em|not}} be a complete metric space{{sfn|Narici|Beckenstein|2011|pp=47-51}} (see this footnote<ref group=note>The [[normed space]] <math>(\R,|\cdot |)</math> is a Banach space where the absolute value is a [[Norm (mathematics)|norm]] on the real line <math>\R</math> that induces the usual [[Euclidean topology]] on <math>\R.</math> Define a metric <math>D : \R \times \R \to \R</math> on <math>\R</math> by <math>D(x, y) =|\arctan(x) - \arctan(y)|</math> for all <math>x, y \in \R.</math> Just like {{nowrap|<math>|\cdot|</math>{{hsp}}'s}} induced metric, the metric <math>D</math> also induces the usual Euclidean topology on <math>\R.</math> However, <math>D</math> is not a complete metric because the sequence <math>x_{\bull} = (x_i)_{i=1}^{\infty}</math> defined by <math>x_i := i</math> is a [[Cauchy sequence|{{nowrap|<math>D</math>-Cauchy}} sequence]] but it does not converge to any point of <math>\R.</math> As a consequence of not converging, this {{nowrap|<math>D</math>-Cauchy}} sequence cannot be a Cauchy sequence in <math>(\R,|\cdot |)</math> (that is, it is not a Cauchy sequence with respect to the norm <math>|\cdot|</math>) because if it was {{nowrap|<math>|\cdot|</math>-Cauchy,}} then the fact that <math>(\R,|\cdot |)</math> is a Banach space would imply that it converges (a contradiction).{{harvnb|Narici|Beckenstein|2011|pp=47–51}}</ref> for an example). In contrast, a theorem of Klee,{{sfn|Schaefer|Wolff|1999|p=35}}<ref name="Klee Inv metrics">{{Cite journal|last1=Klee|first1=V. L.|title=Invariant metrics in groups (solution of a problem of Banach)|year=1952|journal=Proc. Amer. Math. Soc.|volume=3|issue=3|pages=484–487|url=https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf |archive-date=2022-10-09 |url-status=live|doi=10.1090/s0002-9939-1952-0047250-4|doi-access=free}}</ref><ref group=note>The statement of the theorem is: Let <math>d</math> be {{em|any}} metric on a vector space <math>X</math> such that the topology <math>\tau</math> induced by <math>d</math> on <math>X</math> makes <math>(X, \tau)</math> into a topological vector space. If <math>(X, d)</math> is a [[complete metric space]] then <math>(X, \tau)</math> is a [[complete topological vector space]].</ref> which also applies to all [[metrizable topological vector space]]s, implies that if there exists {{em|any}}<ref group=note>This metric <math>D</math> is {{em|not}} assumed to be translation-invariant. So in particular, this metric <math>D</math> does {{em|not}} even have to be induced by a norm.</ref> complete metric <math>D</math> on <math>X</math> that induces the norm topology <math>\tau</math> on <math>X,</math> then <math>(X, \|{\cdot}\|)</math> is a Banach space.

A [[Fréchet space]] is a [[locally convex topological vector space]] whose topology is induced by some translation-invariant complete metric. 
Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the [[space of real sequences]] <math display=inline>\R^{\N} = \prod_{i \in \N} \R</math> with the [[product topology]]). 
However, the topology of every Fréchet space is induced by some [[Countable set|countable]] family of real-valued (necessarily continuous) maps called [[seminorm]]s, which are generalizations of [[Norm (mathematics)|norm]]s. 
It is even possible for a Fréchet space to have a topology that is induced by a countable family of {{em|norms}} (such norms would necessarily be continuous)<ref group=note name=CharacterizationOfContinuityOfANorm>A norm (or [[seminorm]]) <math>p</math> on a topological vector space <math>(X, \tau)</math> is continuous if and only if the topology <math>\tau_p</math> that <math>p</math> induces on <math>X</math> is [[Comparison of topologies|coarser]] than <math>\tau</math> (meaning, <math>\tau_p \subseteq \tau</math>), which happens if and only if there exists some open ball <math>B</math> in <math>(X, p)</math> (such as maybe <math>\{x \in X \mid p(x) < 1\}</math> for example) that is open in <math>(X, \tau).</math></ref>{{sfn|Trèves|2006|pp=57–69}} 
but to not be a Banach/[[normable space]] because 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]].