Difference between revisions of "Topological vector space"

Revision as of 13:52, 16 February 2017

Definition

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

• $(X,\mathcal{J},\mathbb{K})$^{[Note 1]} a topological vector space if it satisfies the following two properties^[1][2]:
  1. $\mathcal{A}:X\times X\rightarrow X$ given by $\mathcal{A}:(u,v)\mapsto u+v$ is continuous - often said simply as "addition is continuous".
  2. $\mathcal{M}:\mathbb{K}\times X\rightarrow X$ given by $\mathcal{M}:(\lambda, x)\mapsto \lambda x$ is also continuous, likewise also often said simply as "multiplication is continuous"
     • Caveat:This is where the definition really matters as it relates the usual topology of the complex numbers (with $\mathbb{R}$'s topology being the same as the subspace topology of this) and the topology we imbue on $X$.

Properties

• For a vector subspace of a topological vector space if there exists an open set contained in the subspace then the spaces are equal
  • Symbolically, if $(X,\mathcal{J},\mathbb{K})$ be a TVS and let $(Y,\mathbb{K})$ be a sub-vector space of $X$ then:
    • $(\exists U\in\mathcal{J}[U\subseteq Y])\implies X\eq Y$

Examples

• $\mathbb{R}^n$ is a topological vector space
• Example:A vector space that is not topological

See also

• Topological group

Notes

1. Jump up ↑ 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 $X$ than the field of a vector space is, so often we will just write:
   • Let $(X,\mathbb{K})$ be a topological vector space

References

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