Editing Topological vector space (section)

===Finest and coarsest vector topology===

Let <math>X</math> be a real or complex vector space.

'''Trivial topology'''

The '''[[trivial topology]]''' or '''indiscrete topology''' <math>\{X, \varnothing\}</math> is always a TVS topology on any vector space <math>X</math> and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on <math>X</math> always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus [[locally compact]]) [[Complete topological vector space|complete]] [[Metrizable topological vector space|pseudometrizable]] [[Seminormed space|seminormable]] [[Locally convex topological vector space|locally convex]] topological vector space. It is [[Hausdorff space|Hausdorff]] if and only if <math>\dim X = 0.</math>

'''Finest vector topology'''

There exists a TVS topology <math>\tau_f</math> on <math>X,</math> called the '''{{visible anchor|finest vector topology}}''' on <math>X,</math> that is finer than every other TVS-topology on <math>X</math> (that is, any TVS-topology on <math>X</math> is necessarily a subset of <math>\tau_f</math>).<ref>{{Cite web|date=2016-04-22|title=A quick application of the closed graph theorem|url=https://terrytao.wordpress.com/2016/04/22/a-quick-application-of-the-closed-graph-theorem/|access-date=2020-10-07| website=What's new| language=en}}</ref>{{sfn|Narici|Beckenstein|2011|p=111}} Every linear map from <math>\left(X, \tau_f\right)</math> into another TVS is necessarily continuous. If <math>X</math> has an uncountable [[Hamel basis]] then <math>\tau_f</math> is {{em|not}} [[Locally convex topological vector space|locally convex]] and {{em|not}} [[Metrizable topological vector space|metrizable]].{{sfn|Narici|Beckenstein|2011|p=111}}