Topological vector space

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Find out if the space must be real, although it looks like it must be. Also find another reference. Demote to B once fleshed out.

Definition

Let [ilmath](X,\mathbb{K})[/ilmath] be a vector space over the field of either the reals, so [ilmath]\mathbb{K}:\eq\mathbb{R} [/ilmath], or the complex numbers, so [ilmath]\mathbb{K}:\eq\mathbb{C} [/ilmath] and let [ilmath]\mathcal{J} [/ilmath] be a topology on [ilmath]X[/ilmath] so that [ilmath](X,\mathcal{ J })[/ilmath] is a topological space. We call the tuple:

• [ilmath](X,\mathcal{J},\mathbb{K})[/ilmath][Note 1] a topological vector space if it satisfies the following two properties[1][2]:
1. [ilmath]\mathcal{A}:X\times X\rightarrow X[/ilmath] given by [ilmath]\mathcal{A}:(u,v)\mapsto u+v[/ilmath] is continuous - often said simply as "addition is continuous".
2. [ilmath]\mathcal{M}:\mathbb{K}\times X\rightarrow X[/ilmath] given by [ilmath]\mathcal{M}:(\lambda, x)\mapsto \lambda x[/ilmath] is also continuous, likewise also often said simply as "multiplication is continuous"