Editing Banach space (section)

====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.