Topological vector space

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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.


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"



See also


  1. This tuple doesn't really matter, nor does the order. I have done it this way for it topology first as in "topological vector space". The topology is "more implicit" when we speak of [ilmath]X[/ilmath] than the field of a vector space is, so often we will just write:
    • Let [ilmath](X,\mathbb{K})[/ilmath] be a topological vector space


  1. Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
  2. Advanced Linear Algebra - Steven Roman