Editing Topological vector space (section)

===Defining topologies using strings===

Let <math>X</math> be a vector space and let <math>U_{\bull} = \left(U_i\right)_{i = 1}^{\infty}</math> be a sequence of subsets of <math>X.</math> Each set in the sequence <math>U_{\bull}</math> is called a '''{{visible anchor|knot}}''' of <math>U_{\bull}</math> and for every index <math>i,</math> <math>U_i</math> is called the '''<math>i</math>-th knot''' of <math>U_{\bull}.</math> The set <math>U_1</math> is called the '''beginning''' of <math>U_{\bull}.</math> The sequence <math>U_{\bull}</math> is/is a:{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}}{{sfn|Schechter|1996|pp=721-751}}{{sfn|Narici|Beckenstein|2011|pp=371-423}}

* '''{{visible anchor|Summative}}''' if <math>U_{i+1} + U_{i+1} \subseteq U_i</math> for every index <math>i.</math>
* '''[[Balanced set|Balanced]]''' (resp. '''[[Absorbing set|absorbing]]''', '''closed''',<ref group="note">The topological properties of course also require that <math>X</math> be a TVS.</ref> '''convex''', '''open''', '''[[Symmetric set|symmetric]]''', '''[[Barrelled space|barrelled]]''', '''[[Absolutely convex set|absolutely convex/disked]]''', etc.) if this is true of every <math>U_i.</math>
* '''{{visible anchor|String}}''' if <math>U_{\bull}</math> is summative, absorbing, and balanced.
* '''{{visible anchor|Topological string}}''' or a '''{{visible anchor|neighborhood string}}''' in a TVS <math>X</math> if <math>U_{\bull}</math> is a string and each of its knots is a neighborhood of the origin in <math>X.</math>

<!-------- START: REMOVED DEFINTION --------------
'''Definition''' ('''Ultrabarrel'''/'''suprabarrel'''): A subset of a TVS <math>X</math> is called an '''ultrabarrel''' (resp. '''suprabarrel''') if it is the beginning of some closed string (resp. of some string) in <math>X.</math>
----------- END: REMOVED DEFINTION --------------->
If <math>U</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in a vector space <math>X</math> then the sequence defined by <math>U_i := 2^{1-i} U</math> forms a string beginning with <math>U_1 = U.</math> This is called the '''natural string of <math>U</math>'''{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}} Moreover, if a vector space <math>X</math> has countable dimension then every string contains an [[Absolutely convex set|absolutely convex]] string.

Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued [[subadditive]] functions. These functions can then be used to prove many of the basic properties of topological vector spaces.

{{Math theorem|name=Theorem|note=<math>\R</math>-valued function induced by a string|math_statement=
Let <math>U_{\bull} = \left(U_i\right)_{i=0}^{\infty}</math> be a collection of subsets of a vector space such that <math>0 \in U_i</math> and <math>U_{i+1} + U_{i+1} \subseteq U_i</math> for all <math>i \geq 0.</math> For all <math>u \in U_0,</math> let <math display=block>\mathbb{S}(u) := \left\{n_{\bull} = \left(n_1, \ldots, n_k\right) ~:~ k \geq 1, n_i \geq 0 \text{ for all } i, \text{ and } u \in U_{n_1} + \cdots + U_{n_k}\right\}.</math>

Define <math>f : X \to [0, 1]</math> by <math>f(x) = 1</math> if <math>x \not\in U_0</math> and otherwise let <math display=block>f(x) := \inf_{} \left\{2^{- n_1} + \cdots 2^{- n_k} ~:~ n_{\bull} = \left(n_1, \ldots, n_k\right) \in \mathbb{S}(x)\right\}.</math>

Then <math>f</math> is subadditive (meaning <math>f(x + y) \leq f(x) + f(y)</math> for all <math>x, y \in X</math>) and <math>f = 0</math> on <math display=inline>\bigcap_{i \geq 0} U_i;</math> so in particular, <math>f(0) = 0.</math> If all <math>U_i</math> are [[symmetric set]]s then <math>f(-x) = f(x)</math> and if all <math>U_i</math> are balanced then <math>f(s x) \leq f(x)</math> for all scalars <math>s</math> such that <math>|s| \leq 1</math> and all <math>x \in X.</math> If <math>X</math> is a topological vector space and if all <math>U_i</math> are neighborhoods of the origin then <math>f</math> is continuous, where if in addition <math>X</math> is Hausdorff and <math>U_{\bull}</math> forms a basis of balanced neighborhoods of the origin in <math>X</math> then <math>d(x, y) := f(x - y)</math> is a metric defining the vector topology on <math>X.</math>
<!--- This theorem is true more generally for commutative additive [[topological group]]s. --->
}}
A proof of the above theorem is given in the article on [[Metrizable topological vector space#Additive sequences|metrizable topological vector spaces]].

If <math>U_{\bull} = \left(U_i\right)_{i \in \N}</math> and <math>V_{\bull} = \left(V_i\right)_{i \in \N}</math> are two collections of subsets of a vector space <math>X</math> and if <math>s</math> is a scalar, then by definition:{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}}

* <math>V_{\bull}</math> '''contains''' <math>U_{\bull}</math>: <math>\ U_{\bull} \subseteq V_{\bull}</math> if and only if <math>U_i \subseteq V_i</math> for every index <math>i.</math>
* '''Set of knots''': <math>\ \operatorname{Knots} U_{\bull} := \left\{U_i : i \in \N\right\}.</math>
* '''Kernel''': <math display=inline>\ \ker U_{\bull} := \bigcap_{i \in \N} U_i.</math>
* '''Scalar multiple''': <math>\ s U_{\bull} := \left(s U_i\right)_{i \in \N}.</math>
* '''Sum''': <math>\ U_{\bull} + V_{\bull} := \left(U_i + V_i\right)_{i \in \N}.</math>
* '''Intersection''': <math>\ U_{\bull} \cap V_{\bull} := \left(U_i \cap V_i\right)_{i \in \N}.</math>

If <math>\mathbb{S}</math> is a collection sequences of subsets of <math>X,</math> then <math>\mathbb{S}</math> is said to be '''directed''' ('''downwards''') '''under inclusion''' or simply '''directed downward''' if <math>\mathbb{S}</math> is not empty and for all <math>U_{\bull}, V_{\bull} \in \mathbb{S},</math> there exists some <math>W_{\bull} \in \mathbb{S}</math> such that <math>W_{\bull} \subseteq U_{\bull}</math> and <math>W_{\bull} \subseteq V_{\bull}</math> (said differently, if and only if <math>\mathbb{S}</math> is a [[Filter (set theory)|prefilter]] with respect to the containment <math>\,\subseteq\,</math> defined above).

'''Notation''': Let <math display=inline>\operatorname{Knots} \mathbb{S} := \bigcup_{U_{\bull} \in \mathbb{S}} \operatorname{Knots} U_{\bull}</math> be the set of all knots of all strings in <math>\mathbb{S}.</math>

Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex.

{{Math theorem|name=Theorem{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}}|note=Topology induced by strings|math_statement=If <math>(X, \tau)</math> is a topological vector space then there exists a set <math>\mathbb{S}</math><ref group=proof>This condition is satisfied if <math>\mathbb{S}</math> denotes the set of all topological strings in <math>(X, \tau).</math></ref> of neighborhood strings in <math>X</math> that is directed downward and such that the set of all knots of all strings in <math>\mathbb{S}</math> is a [[neighborhood basis]] at the origin for <math>(X, \tau).</math> Such a collection of strings is said to be {{em|<math>\tau</math> '''fundamental'''}}.

Conversely, if <math>X</math> is a vector space and if <math>\mathbb{S}</math> is a collection of strings in <math>X</math> that is directed downward, then the set <math>\operatorname{Knots} \mathbb{S}</math> of all knots of all strings in <math>\mathbb{S}</math> forms a [[neighborhood basis]] at the origin for a vector topology on <math>X.</math> In this case, this topology is denoted by <math>\tau_\mathbb{S}</math> and it is called the '''topology generated by <math>\mathbb{S}.</math>'''
}}

If <math>\mathbb{S}</math> is the set of all topological strings in a TVS <math>(X, \tau)</math> then <math>\tau_{\mathbb{S}} = \tau.</math>{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}} A Hausdorff TVS is [[Metrizable topological vector space|metrizable]] [[if and only if]] its topology can be induced by a single topological string.{{sfn|Adasch|Ernst|Keim|1978|pp=10-15}}