PlanetMath: closure of a vector subspace is a vector subspace

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closure of a vector subspace is a vector subspace

(Theorem)

Theorem 1 In a topological vector space the closure of a vector subspace is a vector subspace.

Proof. Let

be the topological vector space over $\mathbbmss{F}$ where $\mathbbmss{F}$ is either $\mathbbmss{R}$ or $\mathbbmss{C}$ , let

be a vector subspace in

, and let $\overline{V}$ be the closure of

. To prove that $\overline{V}$ is a vector subspace of

, it suffices to prove that $\overline{V}$ is non-empty, and

$\displaystyle \lambda x + \mu y \in \overline{V}$

whenever $\lambda,\mu \in \mathbbmss{F}$ and $x,y\in \overline{V}$ .

First, as $V\subseteq \overline{V}$ , $\overline{V}$ contains the zero vector, and $\overline{V}$ is non-empty. Suppose $\lambda,\mu,x,y$ are as above. Then there are nets $(x_i)_{i \in I}$ , $(y_j)_{j \in J}$ in converging to , respectively. In a topological vector space, addition and multiplication are continuous operations. It follows that there is a net $(\lambda x_k + \mu y_k)_{k \in K}$ that converges to $\lambda x + \mu y$ .

We have proven that $\lambda x + \mu y \in \overline{V}$ , so $\overline{V}$ is a vector subspace. $\qedsymbol$

"closure of a vector subspace is a vector subspace" is owned by loner. [ full author list (2) | owner history (1) ]

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Attachments:: closure of a vector subspace in a normed space is a vector subspace (Result) by gumau

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Cross-references: converges, operations, continuous, multiplication, addition, nets, zero vector, contains, closure, vector subspace, topological vector space

This is version 5 of closure of a vector subspace is a vector subspace, born on 2005-02-04, modified 2005-03-01.
Object id is 6710, canonical name is ClosureOfAVectorSubspaceIsAVectorSubspace2.
Accessed 1485 times total.

Classification:

AMS MSC:	15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
	46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)
	54A05 (General topology :: Generalities :: Topological spaces and generalizations )

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