Editing Topological vector space (section)

===Local notions===

A subset <math>E</math> of a vector space <math>X</math> is said to be
* '''[[Absorbing set|absorbing]]''' (in <math>X</math>): if for every <math>x \in X,</math> there exists a real <math>r > 0</math> such that <math>c x \in E</math> for any scalar <math>c</math> satisfying <math>|c| \leq r.</math>{{sfn|Rudin|1991|p=6 §1.4}}
* '''[[Balanced set|balanced]]''' or '''circled''': if <math>t E \subseteq E</math> for every scalar <math>|t| \leq 1.</math>{{sfn|Rudin|1991|p=6 §1.4}}
* '''[[Convex set|convex]]''': if <math>t E + (1 - t) E \subseteq E</math> for every real <math>0 \leq t \leq 1.</math>{{sfn|Rudin|1991|p=6 §1.4}}
* a '''[[Absolutely convex set|disk]]''' or '''[[Absolutely convex set|absolutely convex]]''': if <math>E</math> is convex and balanced.
* '''[[Symmetric set|symmetric]]''': if <math>- E \subseteq E,</math> or equivalently, if <math>- E = E.</math>

Every neighborhood of the origin is an [[absorbing set]] and contains an open [[Balanced set|balanced]] neighborhood of <math>0</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}} so every topological vector space has a local base of absorbing and [[balanced set]]s. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of <math>0;</math> if the space is [[locally convex]] then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin.

'''Bounded subsets'''

A subset <math>E</math> of a topological vector space <math>X</math> is '''[[Bounded set (topological vector space)|bounded]]'''{{sfn|Rudin|1991|p=8}} if for every neighborhood <math>V</math> of the origin there exists <math>t</math> such that <math>E \subseteq t V</math>.

The definition of boundedness can be weakened a bit; <math>E</math> is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set.{{sfn|Narici|Beckenstein|2011|pp=155-176}} Also, <math>E</math> is bounded if and only if for every balanced neighborhood <math>V</math> of the origin, there exists <math>t</math> such that <math>E \subseteq t V.</math> Moreover, when <math>X</math> is locally convex, the boundedness can be characterized by [[seminorm]]s: the subset <math>E</math> is bounded if and only if every continuous seminorm <math>p</math> is bounded on <math>E.</math>{{sfn|Rudin|1991|p=27-28 Theorem 1.37}}

Every [[totally bounded]] set is bounded.{{sfn|Narici|Beckenstein|2011|pp=155-176}} If <math>M</math> is a vector subspace of a TVS <math>X,</math> then a subset of <math>M</math> is bounded in <math>M</math> if and only if it is bounded in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=155-176}}