Editing Banach space (section)

===Topology===

The canonical metric <math>d</math> of a normed space <math>(X, \|{\cdot}\|)</math> induces the usual [[metric topology]] <math>\tau_d</math> on <math>X,</math> which is referred to as the ''canonical'' or ''norm induced [[topology]]''. 
Every normed space is automatically assumed to carry this [[Hausdorff space|Hausdorff]] topology, unless indicated otherwise. 
With this topology, every Banach space is a [[Baire space]], although there exist normed spaces that are Baire but not Banach.{{sfn|Wilansky|2013|p=29}} The norm <math>\|{\cdot}\| : X \to \Reals</math> is always a [[continuous function]] with respect to the topology that it induces.

The open and closed balls of radius <math>r > 0</math> centered at a point <math>x \in X</math> are, respectively, the sets 
<math display=block>B_r(x) := \{z \in X \mid \|z - x\| < r\} \qquad \text{ and } \qquad C_r(x) := \{z \in X \mid \|z - x\| \leq r\}.</math> 
Any such ball is a [[Convex set|convex]] and [[Bounded set (topological vector space)|bounded subset]] of <math>X,</math> but a [[Compact space|compact]] ball/[[Neighbourhood (topology)|neighborhood]] exists if and only if <math>X</math> is [[finite-dimensional]].
In particular, no infinite–dimensional normed space can be [[Locally compact space|locally compact]] or have the [[Montel space|Heine–Borel property]]. 
If <math>x_0</math> is a vector and <math>s \neq 0</math> is a scalar, then 
<math display=block>x_0 + s\,B_r(x) = B_{|s| r}(x_0 + s x) \qquad \text{ and } \qquad x_0 + s\,C_r(x) = C_{|s| r}(x_0 + s x).</math> 
Using <math>s = 1</math> shows that the norm-induced topology is [[Translation invariant topology|translation invariant]], which means that for any <math>x \in X</math> and <math>S \subseteq X,</math> the subset <math>S</math> is [[Open set|open]] (respectively, [[Closed set|closed]]) in <math>X</math> if and only if its translation <math>x + S := \{x + s \mid s \in S\}</math> is open (respectively, closed).
Consequently, the norm induced topology is completely determined by any [[Neighbourhood system|neighbourhood basis]] at the origin. Some common neighborhood bases at the origin include
<math display=block>\{B_r(0) \mid r > 0\}, \qquad \{C_r(0) \mid r > 0\}, \qquad \{B_{r_n}(0) \mid n \in \N\}, \qquad \text{ and }  \qquad \{C_{r_n}(0) \mid n \in \N\},</math>
where <math>r_1, r_2, \ldots</math> can be any sequence of positive real numbers that converges to <math>0</math> in <math>\R</math> (common choices are <math>r_n := \tfrac{1}{n}</math> or <math>r_n := 1/2^n</math>). 
So, for example, any open subset <math>U</math> of <math>X</math> can be written as a union 
<math display=block>U = \bigcup_{x \in I} B_{r_x}(x) = \bigcup_{x \in I} x + B_{r_x}(0) = \bigcup_{x \in I} x + r_x\,B_1(0)</math>
indexed by some subset <math>I \subseteq U,</math> where each <math>r_x</math> may be chosen from the aforementioned sequence <math>r_1, r_2, \ldots.</math> (The open balls can also be replaced with closed balls, although the indexing set <math>I</math> and radii <math>r_x</math> may then also need to be replaced). 
Additionally, <math>I</math> can always be chosen to be [[Countable set|countable]] if <math>X</math> is a {{em|[[separable space]]}}, which by definition means that <math>X</math> contains some countable [[Dense set|dense subset]]. 

====Homeomorphism classes of separable Banach spaces====

All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic. 
Every separable infinite–dimensional [[Hilbert space]] is linearly isometrically isomorphic to the separable Hilbert [[ℓ2 space|sequence space <math>\ell^2(\N)</math>]] with its usual norm <math>\|{\cdot}\|_2.</math> 

The [[Anderson–Kadec theorem]] states that every infinite–dimensional separable [[Fréchet space]] is [[Homeomorphism|homeomorphic]] to the [[product space]] <math display=inline>\prod_{i \in \N} \Reals</math> of countably many copies of <math>\Reals</math> (this homeomorphism need not be a [[linear map]]).<ref>{{harvnb|Bessaga|Pełczyński|1975|p=189}}</ref>{{sfn|Anderson|Schori|1969|p=315}} 
Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique [[up to]] a homeomorphism). <!-- and so as with finite–dimensional spaces, any two separable Fréchet spaces (of any dimensions) are homeomorphic if and only if they have the same dimension.<ref group=note>This means that their dimensions are either both finite and equal or else both infinite.</ref>-->
Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including <math>\ell^2(\N).</math> 
In fact, <math>\ell^2(\N)</math> is even [[Homeomorphism|homeomorphic]] to its own [[Unit sphere|unit {{em|sphere}}]] <math>\{x \in \ell^2(\N) \mid \|x\|_2 = 1\},</math> which stands in sharp contrast to finite–dimensional spaces (the [[Euclidean plane]] <math>\Reals^2</math> is not homeomorphic to the [[unit circle]], for instance). 

This pattern in [[homeomorphism class]]es extends to generalizations of [[Metrizable topological space|metrizable]] ([[locally Euclidean]]) [[topological manifold]]s known as {{em|metric [[Banach manifold]]s}}, which are [[metric space]]s that are around every point, [[locally homeomorphic]] to some open subset of a given Banach space (metric [[Hilbert manifold]]s and metric [[Fréchet manifold]]s are defined similarly).{{sfn|Anderson|Schori|1969|p=315}} 
For example, every open subset <math>U</math> of a Banach space <math>X</math> is canonically a metric Banach manifold modeled on <math>X</math> since the [[inclusion map]] <math>U \to X</math> is an [[Open map|open]] [[local homeomorphism]]. 
Using Hilbert space [[microbundle]]s, David Henderson showed{{sfn|Henderson|1969|p=}} in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or [[Fréchet space|Fréchet]]) space can be [[Topological embedding|topologically embedded]] as an [[Open set|{{em|open}} subset]] of <math>\ell^2(\N)</math> and, consequently, also admits a unique [[smooth structure]] making it into a <math>C^\infty</math> [[Hilbert manifold]]. 

====Compact and convex subsets====

There is a compact subset <math>S</math> of <math>\ell^2(\N)</math> whose [[convex hull]] <math>\operatorname{co}(S)</math> is {{em|not}} closed and thus also {{em|not}} compact.<ref group=note name=ExampleCompactButHullIsNotCompact>Let <math>H</math> be the separable [[Hilbert space]] [[ℓ2 space|<math>\ell^2(\N)</math>]] of square-summable sequences with the usual norm <math>\|{\cdot}\|_2,</math> and let <math>e_n = (0, \ldots, 0, 1, 0, \ldots, 0)</math> be the standard [[orthonormal basis]] (that is, each <math>e_n</math> has zeros in every position except for a <math>1</math> in the <math>n</math><sup>th</sup>-position). The closed set <math>S = \{0\} \cup \{\tfrac{1}{n} e_n \mid n = 1, 2, \ldots\}</math> is compact (because it is [[Sequentially compact space|sequentially compact]]) but its convex hull <math>\operatorname{co} S</math> is {{em|not}} a closed set because the point <math display=inline>h := \sum_{n=1}^{\infty} \tfrac{1}{2^n} \tfrac{1}{n} e_n</math> belongs to the closure of <math>\operatorname{co} S</math> in <math>H</math> but <math>h \not\in\operatorname{co} S</math> (since every point <math>z=(z_1,z_2,\ldots) \in \operatorname{co} S</math> is a finite [[convex combination]] of elements of <math>S</math> and so <math>z_n = 0</math> for all but finitely many coordinates, which is not true of <math>h</math>). However, like in all [[Complete topological vector space|complete]] Hausdorff locally convex spaces, the {{em|closed}} convex hull <math>K := \overline{\operatorname{co}} S</math> of this compact subset is compact. The vector subspace <math>X := \operatorname{span} S = \operatorname{span} \{e_1, e_2, \ldots\}</math> is a [[pre-Hilbert space]] when endowed with the substructure that the Hilbert space <math>H</math> induces on it, but <math>X</math> is not complete and <math>h \not\in C := K \cap X</math> (since <math>h \not\in X</math>). The closed convex hull of <math>S</math> in <math>X</math> (here, "closed" means with respect to <math>X,</math> and not to <math>H</math> as before) is equal to <math>K \cap X,</math> which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of a compact subset might {{em|fail}} to be compact (although it will be [[Totally bounded space|precompact/totally bounded]]).</ref>{{sfn|Aliprantis|Border|2006|p=185}} 
However, like in all Banach spaces, the [[Closed convex hull|{{em|closed}} convex hull]] <math>\overline{\operatorname{co}} S</math> of this (and every other) compact subset will be compact.{{sfn|Trèves|2006|p=145}} In a normed space that is not complete then it is in general {{em|not}} guaranteed that <math>\overline{\operatorname{co}} S</math> will be compact whenever <math>S</math> is; an example<ref group=note name=ExampleCompactButHullIsNotCompact /> can even be found in a (non-complete) [[pre-Hilbert space|pre-Hilbert]] vector subspace of <math>\ell^2(\N).</math>

====As a topological vector space====

This norm-induced topology also makes <math>(X, \tau_d)</math> into what is known as a [[topological vector space]] (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS <math>(X, \tau_d)</math> is {{em|only}} a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is {{em|not}} associated with {{em|any}} particular norm or metric (both of which are "[[Forgetful functor|forgotten]]"). This Hausdorff TVS <math>(X, \tau_d)</math> is even [[Locally convex topological vector space|locally convex]] because the set of all open balls centered at the origin forms a [[neighbourhood basis]] at the origin consisting of convex [[Balanced set|balanced]] open sets. This TVS is also {{em|[[Normable space|normable]]}}, which by definition refers to any TVS whose topology is induced by some (possibly unknown) [[Norm (mathematics)|norm]]. Normable TVSs [[Kolmogorov's normability criterion|are characterized by]] being Hausdorff and having a [[Bounded set (topological vector space)|bounded]] [[Convex set|convex]] neighborhood of the origin. 
All Banach spaces are [[barrelled space]]s, which means that every [[Barrelled set|barrel]] is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that the [[Uniform boundedness principle|Banach–Steinhaus theorem]] holds. 

====Comparison of complete metrizable vector topologies====

The [[Open mapping theorem (functional analysis)|open mapping theorem]] implies that when <math>\tau_1</math> and <math>\tau_2</math> are topologies on <math>X</math> that make both <math>(X, \tau_1)</math> and <math>(X, \tau_2)</math> into [[F-space|complete metrizable TVS]]es (for example, Banach or [[Fréchet space]]s), if one topology is [[Comparison of topologies|finer or coarser]] than the other, then they must be equal (that is, if <math>\tau_1 \subseteq \tau_2</math> or <math>\tau_2 \subseteq \tau_1</math> then <math>\tau_1 = \tau_2</math>).{{sfn|Trèves|2006|pp=166–173}}
So, for example, if <math>(X, p)</math> and <math>(X, q)</math> are Banach spaces with topologies <math>\tau_p</math> and <math>\tau_q,</math> and if one of these spaces has some open ball that is also an open subset of the other space (or, equivalently, if one of <math>p : (X, \tau_q) \to \Reals</math> or <math>q : (X, \tau_p) \to \Reals</math> is continuous), then their topologies are identical and the norms <math>p</math> and <math>q</math> are [[Equivalent norm|equivalent]].