Editing Topological vector space (section)

===Cartesian products===

A [[Cartesian product]] of a family of topological vector spaces, when endowed with the [[product topology]], is a topological vector space. Consider for instance the set <math>X</math> of all functions <math>f: \R \to \R</math> where <math>\R</math> carries its usual [[Euclidean topology]]. This set <math>X</math> is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the [[Cartesian product]] <math>\R^\R,,</math> which carries the natural [[product topology]]. With this product topology, <math>X := \R^{\R}</math> becomes a topological vector space whose topology is called {{em|the topology of [[pointwise convergence]] on <math>\R.</math>}} The reason for this name is the following: if <math>\left(f_n\right)_{n=1}^{\infty}</math> is a [[sequence]] (or more generally, a [[Net (mathematics)|net]]) of elements in <math>X</math> and if <math>f \in X</math> then <math>f_n</math> [[limit of a sequence|converges]] to <math>f</math> in <math>X</math> if and only if for every real number <math>x,</math> <math>f_n(x)</math> converges to <math>f(x)</math> in <math>\R.</math>  This TVS is [[Complete topological vector space|complete]], [[Hausdorff space|Hausdorff]], and [[locally convex]] but not [[Metrizable topological vector space|metrizable]] and consequently not [[normable]]; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form <math>\R f := \{r f : r \in \R\}</math> with <math>f \neq 0</math>).