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

## 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

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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.