Editing Topological vector space (section)

===Normed spaces===

Every [[normed vector space]] has a natural [[Normed vector space#Topological structure|topological structure]]: the norm induces a [[Metric space|metric]] and the metric induces a topology.
This is a topological vector space because{{Citation needed|date=April 2024}}:
#The vector addition map <math>\cdot\, + \,\cdot\; : X \times X \to X</math> defined by <math>(x, y) \mapsto x + y</math> is (jointly) continuous with respect to this topology. This follows directly from the [[triangle inequality]] obeyed by the norm.
#The scalar multiplication map <math>\cdot : \mathbb{K} \times X \to X</math> defined by <math>(s, x) \mapsto s \cdot x,</math> where <math>\mathbb{K}</math> is the underlying scalar field of <math>X,</math> is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm.

Thus all [[Banach space]]s and [[Hilbert space]]s are examples of topological vector spaces.