Editing Topological vector space (section)

===Invariance of vector topologies===

One of the most used properties of vector topologies is that every vector topology is {{em|{{visible anchor|translation invariant}}}}:
:for all <math>x_0 \in X,</math> the map <math>X \to X</math> defined by <math>x \mapsto x_0 + x</math> is a [[homeomorphism]], but if <math>x_0 \neq 0</math> then it is not linear and so not a TVS-isomorphism.
Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if <math>s \neq 0</math> then the linear map <math>X \to X</math> defined by <math>x \mapsto s x</math> is a homeomorphism. Using <math>s = -1</math> produces the negation map <math>X \to X</math> defined by <math>x \mapsto - x,</math> which is consequently a linear homeomorphism and thus a TVS-isomorphism.

If <math>x \in X</math> and any subset <math>S \subseteq X,</math> then <math>\operatorname{cl}_X (x + S) = x + \operatorname{cl}_X S</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}} and moreover, if <math>0 \in S</math> then <math>x + S</math> is a [[Neighborhood (topology)|neighborhood]] (resp. open neighborhood, closed neighborhood) of <math>x</math> in <math>X</math> if and only if the same is true of <math>S</math> at the origin.