The closure of a linear subspace of a normed space is a linear subspace

From Maths
Jump to: navigation, search

Statement

Let [ilmath]((X,[/ilmath][ilmath]\mathbb{K} [/ilmath][ilmath]),\Vert\cdot\Vert)[/ilmath] be a normed space and let [ilmath]L\in\mathcal{P}(X)[/ilmath] be a vector subspace of [ilmath]X[/ilmath], then:

  • [ilmath]\overline{L} [/ilmath] is a vector subspace of [ilmath]X[/ilmath] also - here [ilmath]\overline{A} [/ilmath] denotes the closure of [ilmath]A[/ilmath].[Note 1]

Proof

Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Good practice Q.
  • Proof can be found here Alec (talk) 02:48, 7 April 2017 (UTC)

Notes

  1. Recall that a norm induces a metric and metric induces a topology, so we may speak of closed sets and the closure of an arbitrary subset of this normed space.

References

Grade: B
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
Moderately important